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.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.

quant-pharXiv:2607.06876Jul 9, 20260 citations

Certified randomness between mistrustful players

Carl A. Miller, Yaoyun Shi

It is known that if two players achieve a superclassical score at a nonlocal game $G$ , then their outputs are certifiably random - that is, regardless of the strategy used by the players, a third party will not be able to perfectly predict their outputs (even if he were given their inputs). We prove that for any complete-support game $G$ , there is an explicit nonzero function $F_G$ such that if Alice and Bob achieve a superclassical score of $s$ at $G$ , then Bob has a probability of at most $1 - F_G ( s )$ of correctly guessing Alice's output after the game is played. Our result implies that certifying global randomness through such games must necessarily introduce local randomness.It is known that if two players achieve a superclassical score at a nonlocal game $G$, then their outputs are certifiably random - that is, regardless of the strategy used by the players, a third party will not be able to perfectly predict their outputs (even if he were given their inputs). We prove that for any complete-support game $G$, there is an explicit nonzero function $F_G$ such that if Alice and Bob achieve a superclassical score of $s$ at $G$, then Bob has a probability of at most $1 - F_G ( s )$ of correctly guessing Alice's output after the game is played. Our result implies that certifying global randomness through such games must necessarily introduce local randomness.

arXiv:1610.05140Oct 17, 20160 citations

On symmetric nonlocal games

Andris Ambainis, Dmitry Kravchenko, Nikolay Nahimov +2 more

Theoretical Computer ScienceMar 25, 20132 citations

Randomness in nonlocal games between mistrustful players

Carl A. Miller, Yaoyun Shi

If two quantum players at a nonlocal game G achieve a superclassical score, then their measurement outcomes must be at least partially random from the perspective of any third player. This is the basis for device-independent quantum cryptography. In this paper we address a related question: does a superclassical score at G guarantee that one player has created randomness from the perspective of the other player? We show that for complete-support games, the answer is yes: even if the second player is given the first player’s input at the conclusion of the game, he cannot perfectly recover her output. Thus some amount of local randomness (i.e., randomness possessed by only one player) is always obtained when randomness is certified from nonlocal games with quantum strategies. This is in contrast to non-signaling game strategies, which may produce global randomness without any local randomness. We discuss potential implications for cryptographic protocols between mistrustful parties.If two quantum players at a nonlocal game G achieve a superclassical score, then their measurement outcomes must be at least partially random from the perspective of any third player. This is the basis for device-independent quantum cryptography. In this paper we address a related question: does a superclassical score at G guarantee that one player has created randomness from the perspective of the other player? We show that for complete-support games, the answer is yes: even if the second player is given the first player’s input at the conclusion of the game, he cannot perfectly recover her output. Thus some amount of local randomness (i.e., randomness possessed by only one player) is always obtained when randomness is certified from nonlocal games with quantum strategies. This is in contrast to non-signaling game strategies, which may produce global randomness without any local randomness. We discuss potential implications for cryptographic protocols between mistrustful parties.

arXiv:1706.04984Quantum Information and ComputationMay 1, 20177 citations

It is known that if two players achieve a superclassical score at a nonlocal game

G

, then their outputs are certifiably random - that is, regardless of the strategy used by the players, a third party will not be able to perfectly predict their outputs (even if he were given their inputs). We prove that for any complete-support game

G

, there is an explicit nonzero function

F_G

such that if Alice and Bob achieve a superclassical score of

s

G

, then Bob has a probability of at most

1 - F_G ( s )

of correctly guessing Alice's output after the game is played. Our result implies that certifying global randomness through such games must necessarily introduce local randomness.