### Abstract

We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: If the game is played ℓ times in parallel, does the maximum winning probability decay exponentially in ℓ? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form (1 - Θ(ε))^{ℓ} where 1 - ε is the value of the game and ℓ is the number of repetitions. This question is known as the strong parallel repetition question and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does not hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact perfect) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form (1-Θ(ε^{2})) ^{ℓ}. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.

Original language | English (US) |
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Title of host publication | Proceedings - 25th Annual IEEE Conference on Computational Complexity, CCC 2010 |

Pages | 7-15 |

Number of pages | 9 |

DOIs | |

State | Published - Aug 10 2010 |

Event | 25th Annual IEEE Conference on Computational Complexity, CCC 2010 - Cambridge, MA, United States Duration: Jun 9 2010 → Jun 11 2010 |

### Publication series

Name | Proceedings of the Annual IEEE Conference on Computational Complexity |
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ISSN (Print) | 1093-0159 |

### Other

Other | 25th Annual IEEE Conference on Computational Complexity, CCC 2010 |
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Country | United States |

City | Cambridge, MA |

Period | 6/9/10 → 6/11/10 |

### Keywords

- Entangled two-prover games
- Parallel repetition
- Unique games

### ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Computational Mathematics

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## Cite this

*Proceedings - 25th Annual IEEE Conference on Computational Complexity, CCC 2010*(pp. 7-15). [5497863] (Proceedings of the Annual IEEE Conference on Computational Complexity). https://doi.org/10.1109/CCC.2010.10