[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81976-en":3,"doc-seo-81976-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81976,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","XOR Games at Full Tilt The Hardness of Binary Nonlocal Games","XOR nonlocal games are known to admit efficient polynomial-time approximation of quantum value, and this work studies a variant called tilted XOR games where the winning condition additionally depends on only one output bit. The paper shows a sharp complexity jump: approximating the quantum value of tilted XOR games to constant precision is RE-complete, extending to succinct variants with polynomial-length questions. For classical strategies, tilted and XOR games are equivalent in approximability, while for quantum strategies the hardness is proven via reductions from linear system games.","arXiv :2607 .06876v1 [ quant-ph] 8 Jul 2026  \nXOR Games at Full Tilt The Hardness of Binary Nonlocal Games  \nRichard Cleve * Eric Culf † Aviv Taller ‡  \nAbstract  \nIt is well known that the quantum value of an XOR nonlocal game, where the winning condition depends only on the XOR of the two players’ output bits, may be approximatedin polynomial time. We study a variant of the XOR game model, which we call tilted XOR games, where the winning condition can additionally depend on only one of the output bits. We show that this dramatically increases the expressive power: the computational complexity of the problem of approximating the quantum value of tilted XOR games to constant precision is RE-complete. Also, our result extends to succinct versions of tilted XOR games, where the questions can be polynomial-length binary strings, generated by a polynomial-time verifier.  \nFor classical strategies, the distinction between XOR games and tilted XOR games is inconsequential. Håstad (J. ACM, 2001) shows that they are both NP-complete to approximate, by using a reduction from linear systems to XOR games. Our approach is to show that this is also quantum-sound, but as a reduction from linear system games to tilted XOR games.  \nSince titled XOR games are a special case of binary games (where each party outputs a single bit), our result implies that binary games are RE-hard to approximate.  \n*Institute for Quantum Computing and Cheriton School of Computer Science, University of Waterloo.  \n†Institute for Quantum Computing and Department of Applied Mathematics, University of Waterloo.  \n‡École Polytechnique Fédérale de Lausanne. Part of this work was carried out while at the Weizmann Institute of Science.  \nContents  \n1 Introduction 3  \n1.1 Summary of results .................................. 4  \n2 Preliminaries 6  \n2.1 Notation ........................................ 6  \n2.2 Nonlocal games .................................... 7  \n2.3 XOR games ...................................... 8  \n2.4 Linear constraint system (E3-LIN) games ...................... 10  \n2.5 Håstad’s reduction from E3-LIN games to XOR games ............... 12  \n3 Tilted XOR games 14  \n3.1 Definitions and basic results ............................. 14  \n3.2 The tilted cube game ................................. 16  \n4 Hardness of tilted XOR games 18  \n4.1 Proof of hardness of ungapped version of tilted XOR games ............ 19  \n4.2 Proof of hardness of gapped version of tilted XOR games .............. 20  \n4.2.1 Improved XOR game structure theorem ................... 20  \n4.2.2 Proof of the reduction ............................ 24  \n5 Hardness as a function of completeness and soundness values 27  \n6 Extensions to other models 29  \n6.1 Tilted XOR games with only one instance of a non-XOR question ......... 29  \n6.2 Commuting-operator strategies for tilted XOR games ................ 30  \n6.3 Oracularised games and oracularisable strategies .................. 31  \n6.3.1 Obstacle for oracularised version of tilted XOR games ........... 32  \n6.3.2 Quantum oracularisible strategies for tilted XOR games .......... 35  \n6.4 Tracial strategies for tilted XOR games and noncommutative Max-Cut ....... 38  \n1 Introduction  \nIn a nonlocal game, two physically separated parties (or players), Alice and Bob, are given questions sampled from some fixed probability distribution and are required to produce answers without any communication between them. The parties are said to win the nonlocal game if they satisfy a predetermined predicate depending on both the questions and answers.  \nThe classical value of a nonlocal game is the maximal success probability achievable by a strategy that is restricted to using correlations permitted by classical physics; the quantum value is the supremal success probability attainable by strategies that can utilise pre-shared entanglement, accessing the stronger correlations permitted by quantum physics. Herein, we assume the tensor product model of e","cbCaiouFECBoH9bL","https://ap.wps.com/l/cbCaiouFECBoH9bL","pdf",491005,7,1,44,"English","en",105,"# Introduction\n## Summary of results\n# Preliminaries\n## Nonlocal games\n## XOR games\n## Linear constraint system (E3-LIN) games\n## Håstad’s reduction from E3-LIN games to XOR games\n# Tilted XOR games\n## Definitions and basic results\n## The tilted cube game\n# Hardness of tilted XOR games\n## Hardness of ungapped version\n## Hardness of gapped version\n# Hardness as a function of completeness and soundness values\n# Extensions to other models\n## Tilted XOR games with only one instance of a non-XOR question\n## Commuting-operator strategies\n## Oracularised games and oracularisable strategies\n## Tracial strategies","[{\"question\":\"What are tilted XOR games, and how do they differ from XOR nonlocal games?\",\"answer\":\"Tilted XOR games modify the XOR-game winning condition by allowing dependence on one additional output bit. The winning predicate thus depends on a restricted structure beyond merely the XOR of output bits.\"},{\"question\":\"What complexity result does the paper prove for approximating tilted XOR games’ quantum value?\",\"answer\":\"Approximating the quantum value of tilted XOR games to constant precision is shown to be RE-complete. This hardness persists in succinct versions where questions are polynomial-length binary strings generated by a polynomial-time verifier.\"},{\"question\":\"How does the paper relate classical and quantum strategies for tilted XOR games?\",\"answer\":\"For classical strategies, the distinction between XOR and tilted XOR games does not affect approximability: both become NP-complete to approximate by reductions previously known from linear systems to XOR games. The paper then establishes that, in contrast, the quantum version remains hard via a quantum-sound reduction from linear system games to tilted XOR games.\"}]",1784177366,111,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"xor-games-at-full-tilt-the-hardness-of-binary-nonlocal-games","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/xor-games-at-full-tilt-the-hardness-of-binary-nonlocal-games/81976/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What are tilted XOR games, and how do they differ from XOR nonlocal games?","Question",{"text":76,"@type":77},"Tilted XOR games modify the XOR-game winning condition by allowing dependence on one additional output bit. The winning predicate thus depends on a restricted structure beyond merely the XOR of output bits.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What complexity result does the paper prove for approximating tilted XOR games’ quantum value?",{"text":81,"@type":77},"Approximating the quantum value of tilted XOR games to constant precision is shown to be RE-complete. This hardness persists in succinct versions where questions are polynomial-length binary strings generated by a polynomial-time verifier.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the paper relate classical and quantum strategies for tilted XOR games?",{"text":85,"@type":77},"For classical strategies, the distinction between XOR and tilted XOR games does not affect approximability: both become NP-complete to approximate by reductions previously known from linear systems to XOR games. The paper then establishes that, in contrast, the quantum version remains hard via a quantum-sound reduction from linear system games to tilted XOR games.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":46,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]