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Improved Parallel Repetition for GHZ-Supported Games via Spreadness

Authors Yang P. Liu , Shachar Lovett , Kunal Mittal

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Parallel Repetition
  • GHZ Game
  • Algebraic Spreadness

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Abstract

We prove that for any 3-player game G, whose query distribution has the same support as the GHZ game (i.e., all x,y,z ∈ {0,1} satisfying x+y+z = 0 (mod 2)), the value of the n-fold parallel repetition of G decays exponentially fast: 

val(G^{⊗ n}) ≤ exp(-n^c) 

for all sufficiently large n, where c > 0 is an absolute constant. 
We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant ε > 0, the probability that the players win at least a (3/4+ε) fraction of the n coordinates is at most exp(-n^c), where c = c(ε) > 0 is a constant.
In both settings, our work exponentially improves upon the previous best known bounds which were only polynomially small, i.e., of the order n^{-Ω(1)}. Our key technical tool is the notion of algebraic spreadness adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.

Cite As Get BibTex

Yang P. Liu, Shachar Lovett, and Kunal Mittal. Improved Parallel Repetition for GHZ-Supported Games via Spreadness. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 6:1-6:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.6

Author Details

Yang P. Liu
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Shachar Lovett
  • Department of Computer Science and Engineering, University of California, San Diego, CA, USA
Kunal Mittal
  • Department of Computer Science, Courant Institute of Mathematical Sciences, New York University, NY, USA

Funding

  • Lovett, Shachar: Supported by Simons Investigator Award #929894 and NSF award CCF-2425349.
  • Mittal, Kunal: Research supported a Simons Investigator Award.

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