XOR Games at Full Tilt: The Hardness of Binary Nonlocal Games

2026-07-07

AI Summary

This work investigates the computational complexity of approximating the quantum value of tilted XOR games. By reducing linear system games to tilted XOR games, the authors establish that approximating their quantum value to constant precision is RE-complete, a result that extends to succinctly represented instances. This constitutes the first proof of RE-completeness for tilted XOR games under quantum strategies, thereby overcoming the limitation that classical XOR games admit polynomial-time approximation algorithms. The result further implies RE-hardness for general two-player nonlocal games, significantly advancing our understanding of the computational boundaries of quantum nonlocality.

Abstract

It 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 approximated in 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. For 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. Since 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.

Problem

• tilted XOR games
• quantum value
• computational complexity
• binary nonlocal games
• RE-completeness

Innovation

• tilted XOR games
• quantum value approximation
• RE-completeness
• binary nonlocal games
• succinct representation