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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Securing Smart Manufacturing: Detection of Cyber-Physical Attacks in CNC-Based Systems

As Industry 4.0 advances, the integration of computer numerical control (CNC) machines and advanced manufacturing technologies is transforming production into smart manufacturing systems that blend physical and digital processes as cyber-physical systems. However, this increased cyber-physical connectivity exposes manufacturing systems to cyber threats that can cause severe operational and financial disruptions. This paper presents a comparative study on cyber attacks and anomaly detection techniques in manufacturing, focusing on network traffic from CNC machines. The data extracted from network packets includes machine commands and control signals exchanged between the machine's interface and control system, crucial for maintaining operational integrity. We explore two types of cyber attacks, design modification and command injection, which pose substantial risks to CNC machine productivity and system integrity. Our investigation involves experiments on a real CNC system, highlighting the urgent need for effective detection mechanisms. To address these threats, we evaluate three anomaly detection methods: dynamic time warping (DTW), rolling average, and a deep learning, long short-term memory (LSTM) time-series-based autoencoder. Each is assessed for its effectiveness in identifying anomalous behaviors caused by the attacks. Our findings demonstrate the unique strengths and limitations of each detection technique, providing a deeper understanding of their applicability in realworld manufacturing environments. The comparative analysis indicates that while certain methods are highly effective against specific attack types, others offer broader applicability across different attacks. This study contributes to the accurate detection of anomalies in CNC machining processes, thereby enhancing the reliability and security of smart manufacturing systems against diverse cyber threats.

Williams, Bethanie [Tennessee Technological Univer↗
NASA NTRS2017

MiniWall Tool for Analyzing CFD and Wind Tunnel Large Data Sets

It is challenging to review and assimilate large data sets created by Computational Fluid Dynamics (CFD) simulations and wind tunnel tests. Over the past 10 years, NASA Ames Research Center has developed and refined a software tool dubbed the MiniWall to increase productivity in reviewing and understanding large CFD-generated data sets. Under the recent NASA ERA project, the application of the tool expanded to enable rapid comparison of experimental and computational data. The MiniWall software is browser based so that it runs on any computer or device that can display a web page. It can also be used remotely and securely by using web server software such as the Apache HTTP server. The MiniWall software has recently been rewritten and enhanced to make it even easier for analysts to review large data sets and extract knowledge and understanding from these data sets. This paper describes the MiniWall software and demonstrates how the different features are used to review and assimilate large data sets.

Computational Fluid Dyanmics↗
NASA NTRS2017

MiniWall Tool for Analyzing CFD and Wind Tunnel Large Data Sets

It is challenging to review and assimilate large data sets created by Computational Fluid Dynamics (CFD) simulations and wind tunnel tests. Over the past 10 years, NASA Ames Research Center has developed and refined a software tool dubbed the "MiniWall" to increase productivity in reviewing and understanding large CFD‐generated data sets. Under the recent NASA ERA project, the application of the tool expanded to enable rapid comparison of experimental and computational data. The MiniWall software is browser based so that it runs on any computer or device that can display a web page. It can also be used remotely and securely by using web server software such as the Apache HTTP Server. The MiniWall software has recently been rewritten and enhanced to make it even easier for analysts to review large data sets and extract knowledge and understanding from these data sets. This paper describes the MiniWall software and demonstrates how the different features are used to review and assimilate large data sets.

Data Analysis↗
DOE OSTI2026

Regression Analysis with the Directed Infusion of Data

Integrating artificial intelligence and machine learning tools into industry necessitates large-scale collaborative efforts that ensure the robust and accurate execution of downstream analytics such as time series prediction, uncertainty quantification, grid optimization, and condition monitoring. However, concerns related to data privacy pervade the nuclear industry due to the proprietary nature of its data and the possibility of data leakage. Legacy techniques such as encryption often require the explicit transmission of data to trustworthy parties, thereby inviting data leakage concerns. The ideal collaboration scenario avoids the explicit dissemination of data/code while maintaining experimental fidelity, which is currently accomplished using various techniques such as trusted execution environments, homomorphic encryption, differential privacy, and multimatrix masking. These techniques, however, often necessitate a trade-off between trust, efficiency, and utility. This article extends a previously proposed technique called the directed infusion of data (DIOD) that ensures data privacy, allows for scalable obfuscation, and combats the risk of data leakage without compromising utility. The experiments discussed in this article examine a regression-type scenario using DIOD with the goal of preserving the inferential link between two variables. Using the point-kinetics equations, regression experiments compare the performance of a model trained using the original data to that of a model trained using the obfuscated data, which produced identical results. Our claim is further strengthened by an information theoretic proof and experiment, which showed that the inferential content between variables remains the same after obfuscation, thereby avoiding the required communication of the proprietary data.

47 - OTHER INSTRUMENTATION↗
NASA NTRS2011

Redefining Tactical Operations for MER Using Cloud Computing

The Mars Exploration Rover Mission (MER) includes the twin rovers, Spirit and Opportunity, which have been performing geological research and surface exploration since early 2004. The rovers' durability well beyond their original prime mission (90 sols or Martian days) has allowed them to be a valuable platform for scientific research for well over 2000 sols, but as a by-product it has produced new challenges in providing efficient and cost-effective tactical operational planning. An early stage process adaptation was the move to distributed operations as mission scientists returned to their places of work in the summer of 2004, but they would still came together via teleconference and connected software to plan rover activities a few times a week. This distributed model has worked well since, but it requires the purchase, operation, and maintenance of a dedicated infrastructure at the Jet Propulsion Laboratory. This server infrastructure is costly to operate and the periodic nature of its usage (typically heavy usage for 8 hours every 2 days) has made moving to a cloud based tactical infrastructure an extremely tempting proposition. In this paper we will review both past and current implementations of the tactical planning application focusing on remote plan saving and discuss the unique challenges present with long-latency, distributed operations. We then detail the motivations behind our move to cloud based computing services and as well as our system design and implementation. We will discuss security and reliability concerns and how they were addressed

Mars↗
NASA NTRS2010

Anatomy of a Security Operations Center

Many agencies and corporations are either contemplating or in the process of building a cyber Security Operations Center (SOC). Those Agencies that have established SOCs are most likely working on major revisions or enhancements to existing capabilities. As principle developers of the NASA SOC; this Presenters' goals are to provide the GFIRST community with examples of some of the key building blocks of an Agency scale cyber Security Operations Center. This presentation viII include the inputs and outputs, the facilities or shell, as well as the internal components and the processes necessary to maintain the SOC's subsistence - in other words, the anatomy of a SOC. Details to be presented include the SOC architecture and its key components: Tier 1 Call Center, data entry, and incident triage; Tier 2 monitoring, incident handling and tracking; Tier 3 computer forensics, malware analysis, and reverse engineering; Incident Management System; Threat Management System; SOC Portal; Log Aggregation and Security Incident Management (SIM) systems; flow monitoring; IDS; etc. Specific processes and methodologies discussed include Incident States and associated Work Elements; the Incident Management Workflow Process; Cyber Threat Risk Assessment methodology; and Incident Taxonomy. The Evolution of the Cyber Security Operations Center viII be discussed; starting from reactive, to proactive, and finally to proactive. Finally, the resources necessary to establish an Agency scale SOC as well as the lessons learned in the process of standing up a SOC viII be presented.

Wang, John↗
NASA NTRS2007

System for Secure Integration of Aviation Data

The Aviation Data Integration System (ADIS) of Ames Research Center has been established to promote analysis of aviation data by airlines and other interested users for purposes of enhancing the quality (especially safety) of flight operations. The ADIS is a system of computer hardware and software for collecting, integrating, and disseminating aviation data pertaining to flights and specified flight events that involve one or more airline(s). The ADIS is secure in the sense that care is taken to ensure the integrity of sources of collected data and to verify the authorizations of requesters to receive data. Most importantly, the ADIS removes a disincentive to collection and exchange of useful data by providing for automatic removal of information that could be used to identify specific flights and crewmembers. Such information, denoted sensitive information, includes flight data (here signifying data collected by sensors aboard an aircraft during flight), weather data for a specified route on a specified date, date and time, and any other information traceable to a specific flight. The removal of information that could be used to perform such tracing is called "deidentification." Airlines are often reluctant to keep flight data in identifiable form because of concerns about loss of anonymity. Hence, one of the things needed to promote retention and analysis of aviation data is an automated means of de-identification of archived flight data to enable integration of flight data with non-flight aviation data while preserving anonymity. Preferably, such an automated means would enable end users of the data to continue to use pre-existing data-analysis software to identify anomalies in flight data without identifying a specific anomalous flight. It would then also be possible to perform statistical analyses of integrated data. These needs are satisfied by the ADIS, which enables an end user to request aviation data associated with de-identified flight data. The ADIS includes client software integrated with other software running on flight-operations quality-assurance (FOQA) computers for purposes of analyzing data to study specified types of events or exceedences (departures of flight parameters from normal ranges). In addition to ADIS client software, ADIS includes server hardware and software that provide services to the ADIS clients via the Internet (see figure). The ADIS server receives and integrates flight and non-flight data pertaining to flights from multiple sources. The server accepts data updates from authorized sources only and responds to requests from authorized users only. In order to satisfy security requirements established by the airlines, (1) an ADIS client must not be accessible from the Internet by an unauthorized user and (2) non-flight data as airport terminal information system (ATIS) and weather data must be displayed without any identifying flight information. ADIS hardware and software architecture as well as encryption and data display scheme are designed to meet these requirements. When a user requests one or more selected aviation data characteristics associated with an event (e.g., a collision, near miss, equipment malfunction, or exceedence), the ADIS client augments the request with date and time information from encrypted files and submits the augmented request to the server. Once the user s authorization has been verified, the server returns the requested information in de-identified form.

Kulkarni, Deepak↗
DOE OSTI2024

Integration of the NCRC Database and Other INL Databases

The Nuclear Computational Resource Center provides a portal by which industry professionals, educational staff, students, national laboratory employees, and others may request access to certain engineering software tools. As the tools provided through the Nuclear Computational Resource Center portal are not open-source and freely available, a set of approvals are necessary before access is granted. All code recipients must be associated with an institution that has a license with Idaho National Laboratory for the code requested. Information about these licenses is controlled by Idaho National Laboratory’s Technology Deployment organization and housed in a Technology Deployment database. Those requesting code access who are not citizens of the United States must also have a security plan, mandated by Idaho National Laboratory policy. Security plans are managed by the International Access Program and are stored in an International Access Program database known as IFacts. Granting access to software thus depends on information stored in the Technology Deployment database and IFacts. In the past, no connection between the Nuclear Computational Resource Center portal and these databases existed, making checking the status of license agreements and security plans time consuming and error prone. This report demonstrates that the Nuclear Computational Resource Center portal now connects to both the Technology Deployment database and IFacts, greatly improving the ease of use of the Nuclear Computational Resource Center system for administrators, which leads to a better overall experience for those requesting code access.

99 GENERAL AND MISCELLANEOUS↗
DOE OSTI2024

Summer 2024 INL Intern Poster Session Submission - Brian Schumitz

This LRS submission is my poster for the INL Intern Poster Session, Summer 2024. Abstract: The Software Engineering and Cybersecurity Lab (SECL) at Montana State University has developed PIQUE, a system for evaluating software quality. PIQUE's adaptability allows for language-specific static-analysis operations, including a model for assessing cloud microservice ecosystems. These ecosystems often rely on Docker for efficient deployment and management of containerized services. Our research focuses on evaluating the network quality within these microservice ecosystems. To automate this process, we're utilizing Snort, an open-source intrusion detection system renowned for its ability to detect and log network traffic. By leveraging Snort's customizable rules, we aim to construct comprehensive testing methods for measuring and quantifying the network quality based on traffic between Docker containers. This research aims to enhance the overall security and reliability of cloud microservice ecosystems by providing automated and robust quality evaluation mechanisms, ultimately contributing to the advancement of software engineering practices in these environments

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2012

The Namibia Early Flood Warning System, A CEOS Pilot Project

Over the past year few years, an international collaboration has developed a pilot project under the auspices of Committee on Earth Observation Satellite (CEOS) Disasters team. The overall team consists of civilian satellite agencies. For this pilot effort, the development team consists of NASA, Canadian Space Agency, Univ. of Maryland, Univ. of Colorado, Univ. of Oklahoma, Ukraine Space Research Institute and Joint Research Center(JRC) for European Commission. This development team collaborates with regional , national and international agencies to deliver end-to-end disaster coverage. In particular, the team in collaborating on this effort with the Namibia Department of Hydrology to begin in Namibia . However, the ultimate goal is to expand the functionality to provide early warning over the South Africa region. The initial collaboration was initiated by United Nations Office of Outer Space Affairs and CEOS Working Group for Information Systems and Services (WGISS). The initial driver was to demonstrate international interoperability using various space agency sensors and models along with regional in-situ ground sensors. In 2010, the team created a preliminary semi-manual system to demonstrate moving and combining key data streams and delivering the data to the Namibia Department of Hydrology during their flood season which typically is January through April. In this pilot, a variety of moderate resolution and high resolution satellite flood imagery was rapidly delivered and used in conjunction with flood predictive models in Namibia. This was collected in conjunction with ground measurements and was used to examine how to create a customized flood early warning system. During the first year, the team made use of SensorWeb technology to gather various sensor data which was used to monitor flood waves traveling down basins originating in Angola, but eventually flooding villages in Namibia. The team made use of standardized interfaces such as those articulated under the Open Cloud Consortium (OGC) Sensor Web Enablement (SWE) set of web services was good [1][2]. However, it was discovered that in order to make a system like this functional, there were many performance issues. Data sets were large and located in a variety of location behind firewalls and had to be accessed across open networks, so security was an issue. Furthermore, the network access acted as bottleneck to transfer map products to where they are needed. Finally, during disasters, many users and computer processes act in parallel and thus it was very easy to overload the single string of computers stitched together in a virtual system that was initially developed. To address some of these performance issues, the team partnered with the Open Cloud Consortium (OCC) who supplied a Computation Cloud located at the University of Illinois at Chicago and some manpower to administer this Cloud. The Flood SensorWeb [3] system was interfaced to the Cloud to provide a high performance user interface and product development engine. Figure 1 shows the functional diagram of the Flood SensorWeb. Figure 2 shows some of the functionality of the Computation Cloud that was integrated. A significant portion of the original system was ported to the Cloud and during the past year, technical issues were resolved which included web access to the Cloud, security over the open Internet, beginning experiments on how to handle surge capacity by using the virtual machines in the cloud in parallel, using tiling techniques to render large data sets as layers on map, interfaces to allow user to customize the data processing/product chain and other performance enhancing techniques. The conclusion reached from the effort and this presentation is that defining the interoperability standards in a small fraction of the work. For example, once open web service standards were defined, many users could not make use of the standards due to security restrictions. Furthermore, once an interoperable sysm is functional, then a surge of users can render a system unusable, especially in the disaster domain.

Mandl, Daniel↗
DOE OSTI2026

Transportation Secure Data Center: Frequently Asked Questions for Data Owners/Contributors

The Transportation Secure Data Center is a centralized repository for detailed transportation data from travel and transit surveys and studies conducted across the nation. It makes vital transportation data broadly available to users while preserving the privacy of survey participants. Hundreds of datasets from surveys and studies of household travel and transit passenger travel are archived in the TSDC, including surveys and studies conducted by state departments of transportation, metropolitan planning organizations, transit agencies, cities, and other public agencies. Detailed data from travel surveys and studies are extremely valuable for research purposes. However, the fine-grained information they contain could potentially be misused to identify individual travelers, so access to these data should only be granted with safeguards in place to protect participant privacy. The TSDC was created to address this challenge and to relieve public agencies from the burden of archiving their data and responding to data requests.

33 ADVANCED PROPULSION SYSTEMS↗
NASA NTRS2002

Accessing Wind Tunnels From NASA's Information Power Grid

The NASA Ames wind tunnel customers are one of the first users of the Information Power Grid (IPG) storage system at the NASA Advanced Supercomputing Division. We wanted to be able to store their data on the IPG so that it could be accessed remotely in a secure but timely fashion. In addition, incorporation into the IPG allows future use of grid computational resources, e.g., for post-processing of data, or to do side-by-side CFD validation. In this paper, we describe the integration of grid data access mechanisms with the existing DARWIN web-based system that is used to access wind tunnel test data. We also show that the combined system has reasonable performance: wind tunnel data may be retrieved at 50Mbits/s over a 100 base T network connected to the IPG storage server.

Becker, Jeff↗
NASA NTRS2010

Support for Systematic Code Reviews with the SCRUB Tool

SCRUB is a code review tool that supports both large, team-based software development efforts (e.g., for mission software) as well as individual tasks. The tool was developed at JPL to support a new, streamlined code review process that combines human-generated review reports with program-generated review reports from a customizable range of state-of-the-art source code analyzers. The leading commercial tools include Codesonar, Coverity, and Klocwork, each of which can achieve a reasonably low rate of false-positives in the warnings that they generate. The time required to analyze code with these tools can vary greatly. In each case, however, the tools produce results that would be difficult to realize with human code inspections alone. There is little overlap in the results produced by the different analyzers, and each analyzer used generally increases the effectiveness of the overall effort. The SCRUB tool allows all reports to be accessed through a single, uniform interface (see figure) that facilitates brows ing code and reports. Improvements over existing software include significant simplification, and leveraging of a range of commercial, static source code analyzers in a single, uniform framework. The tool runs as a small stand-alone application, avoiding the security problems related to tools based on Web browsers. A developer or reviewer, for instance, must have already obtained access rights to a code base before that code can be browsed and reviewed with the SCRUB tool. The tool cannot open any files or folders to which the user does not already have access. This means that the tool does not need to enforce or administer any additional security policies. The analysis results presented through the SCRUB tool s user interface are always computed off-line, given that, especially for larger projects, this computation can take longer than appropriate for interactive tool use. The recommended code review process that is supported by the SCRUB tool consists of three phases: Code Review, Developer Response, and Closeout Resolution. In the Code Review phase, all tool-based analysis reports are generated, and specific comments from expert code reviewers are entered into the SCRUB tool. In the second phase, Developer Response, the developer is asked to respond to each comment and tool-report that was produced, either agreeing or disagreeing to provide a fix that addresses the issue that was raised. In the third phase, Closeout Resolution, all disagreements are discussed in a meeting of all parties involved, and a resolution is made for all disagreements. The first two phases generally take one week each, and the third phase is concluded in a single closeout meeting.

Holzmann, Gerald J.↗
NASA NTRS2013

Common Practice Lightning Strike Protection Characterization Technique to Quantify Damage Mechanisms on Composite Substrates

To support FAA certification airworthiness standards, composite substrates are subjected to lightning direct-effect electrical waveforms to determine performance characteristics of the lightning strike protection (LSP) conductive layers used to protect composite substrates. Test results collected from independent LSP studies are often incomparable due to variability in test procedures & applied practices at different organizations, which impairs performance correlations between different LSP data sets. Under a NASA supported contract, The Boeing Company developed technical procedures and documentation as guidance in order to facilitate a test method for conducting universal common practice lightning strike protection test procedures. The procedures obtain conformity in future lightning strike protection evaluations to allow meaningful performance correlations across data sets. This universal common practice guidance provides the manufacturing specifications to fabricate carbon fiber reinforced plastic (CFRP) test panels, including finish, grounding configuration, and acceptable methods for pretest nondestructive inspection (NDI) and posttest destructive inspection. The test operations guidance elaborates on the provisions contained in SAE ARP5416 to address inconsistencies in the generation of damage protection performance data, so as to provide for maximum achievable correlation across capable lab facilities. In addition, the guidance details a direct effects test bed design to aid in quantification of the multi-physical phenomena surrounding a lightning direct attachment supporting validation data requirements for the development of predictive computational modeling. The lightning test bed is designed to accommodate a repeatable installation procedure to secure the test panel and eliminate test installation uncertainty. It also facilitates a means to capture the electrical waveform parameters in 2 dimensions, along with the mechanical displacement and thermal heating parameters which occur during lightning attachment. Following guidance defined in the universal common practice LSP test documents, protected and unprotected CFRP panels were evaluated at 20, 40 and 100KAmps. This report presents analyzed data demonstrating the scientific usefulness of the common practice approach. Descriptions of the common practice CFRP test articles, LSP test bed fixture, and monitoring techniques to capture the electrical, mechanical and thermal parameters during lightning attachment are presented here. Two methods of measuring the electrical currents were evaluated, inductive current probes and a newly developed fiberoptic sensor. Two mechanical displacement methods were also examined, optical laser measurement sensors and a digital imaging correlation camera system. Recommendations are provided to help users implement the common practice test approach and obtain LSP test characterizations comparable across data sets.

Szatkowski, George N.↗
DOE OSTI2024

IRI Technology Landscape – A survey of re-usable components and methodologies

This document describes technical implementation details on network access schemes connecting API-driven workflows to supercomputer centers. API-driven workflows are a central theme in connected computing, since they bring the terminal-mainframe' access pattern present since the 1970s up to the task of interfacing with modern web browser technologies. Both security (HTTPS/TLS/IPSec/VPNs/public key cryptography/digital signatures) and network protocol stacks (HTTP-REST APIs, tokens, gRPC, SRTP) have evolved to the point where implementing API-driven workflows is possible using stable, secure off-the-shelf software.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Selective Recovery of Critical Minerals from Simulated Electronic Wastes Via Reaction‐Diffusion Coupling

Abstract Atom‐ and energy‐efficient chemical separations are urgently needed to meet the surging demand for critical materials that has strained supply chains and threatened environmental damage. In this study, we used reaction‐diffusion coupling to separate iron, neodymium, and dysprosium ions from model feedstocks of permanent magnets, which are typically found in electronic wastes. Feedstock solutions were placed in contact with a hydrogel loaded with potassium hydroxide and/or dibutyl phosphate, resulting in complex precipitation patterns as the various metal ions diffused into the reaction medium. Specifically, we observed the precipitation of up to 40 mM of iron from the feedstock, followed by the enrichment of 73 % dysprosium, and the extraction of >95 % neodymium product at a further distance from the solution‐gel interface. We designed a series of experiments and simulations to determine the relevant ion diffusivities, D Nd =5.4×10 −10 and D Dy =5.1×10 −10 m 2 /s, and precipitation rates, k Nd =1.0×10 −5 and k Dy =5.0×10 −3 m 9 mol −3 s −1 , which enabled a numerical model to be established for predicting the distribution of products in the reaction medium. Our proof‐of‐concept study validates reaction‐diffusion coupling as an effective and versatile approach for critical materials separations, without relying on ligands, membranes, resins, or other specialty chemicals.

Wang, Qingpu [Physical and Computational Sciences ↗
NASA NTRS1979

Flight design system level C requirements. Solid rocket booster and external tank impact prediction processors

The prediction of the SRB and ET impact areas requires six separate processors. The SRB impact prediction processor computes the impact areas and related trajectory data for each SRB element. Output from this processor is stored on a secure file accessible by the SRB impact plot processor which generates the required plots. Similarly the ET RTLS impact prediction processor and the ET RTLS impact plot processor generates the ET impact footprints for return-to-launch-site (RTLS) profiles. The ET nominal/AOA/ATO impact prediction processor and the ET nominal/AOA/ATO impact plot processor generate the ET impact footprints for non-RTLS profiles. The SRB and ET impact processors compute the size and shape of the impact footprints by tabular lookup in a stored footprint dispersion data base. The location of each footprint is determined by simulating a reference trajectory and computing the reference impact point location. To insure consistency among all flight design system (FDS) users, much input required by these processors will be obtained from the FDS master data base.

Seale, R. H.↗
DOE OSTI2025

Playing Nonlocal Games across a Topological Phase Transition on a Quantum Computer

Many-body quantum games provide a natural perspective on phases of matter in quantum hardware, crisply relating the quantum correlations inherent in phases of matter to the securing of quantum advantage at a device-oriented task. In this Letter, we introduce a family of multiplayer quantum games for which topologically ordered phases of matter are a resource yielding quantum advantage. Unlike previous examples, quantum advantage persists away from the exactly solvable point and is robust to arbitrary local perturbations, irrespective of system size. We demonstrate this robustness experimentally on Quantinuum’s H1-1 quantum computer by playing the game with a continuous family of randomly deformed toric code states that can be created with constant-depth circuits leveraging midcircuit measurements and unitary feedback. We are thus able to tune through a topological phase transition—witnessed by the loss of robust quantum advantage—on currently available quantum hardware. This behavior is contrasted with an analogous family of deformed Greenberger-Horne-Zeilinger states, for which arbitrarily weak local perturbations destroy quantum advantage in the thermodynamic limit. Lastly, we discuss a topological interpretation of the game, which leads to a natural generalization involving an arbitrary number of players.

97 MATHEMATICS AND COMPUTING↗