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Improved Parallel Repetition for GHZ-Supported Games via Spreadness

Yang P. Liu, Shachar Lovett, Kunal Mittal

TL;DR

This paper proves that for any 3-player game with a GHZ-support query distribution, the value under n-fold parallel repetition decays as $\text{val}(\mathcal{G}^{\otimes n}) \le \exp(-n^{c})$ for some absolute $c>0$, and it also establishes a concentration bound around the value for the GHZ distribution. The authors introduce algebraic spreadness, a pseudorandomness concept inspired by Kelley–Meka, and develop a uniformization framework to decompose complex input-sets into spread components and then into uniform square covers. The key technical contribution is showing that distributing inputs via squares hardens coordinates and that conditioned distributions can be approximated by mixtures of spread components, enabling the stretched-exponential decay in the general GHZ-support setting (beyond XOR structure). These results significantly advance understanding of multiplayer parallel repetition, with implications for complexity theory and high-dimensional combinatorics, by moving from polynomial to stretched-exponential decay under a natural, broad query-support condition. The methods also yield a concentration bound for the GHZ game, strengthening prior exponential decay results and offering new tools for analyzing high-dimensional combinatorial games through algebraic pseudorandomness.

Abstract

We prove that for any 3-player game $\mathcal G$, whose query distribution has the same support as the GHZ game (i.e., all $x,y,z\in \{0,1\}$ satisfying $x+y+z=0\pmod{2}$), the value of the $n$-fold parallel repetition of $\mathcal G$ decays exponentially fast: \[ \text{val}(\mathcal G^{\otimes n}) \leq \exp(-n^c)\] 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 $\left(\frac{3}{4}+ε\right)$ 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 \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.

Improved Parallel Repetition for GHZ-Supported Games via Spreadness

TL;DR

This paper proves that for any 3-player game with a GHZ-support query distribution, the value under n-fold parallel repetition decays as for some absolute , and it also establishes a concentration bound around the value for the GHZ distribution. The authors introduce algebraic spreadness, a pseudorandomness concept inspired by Kelley–Meka, and develop a uniformization framework to decompose complex input-sets into spread components and then into uniform square covers. The key technical contribution is showing that distributing inputs via squares hardens coordinates and that conditioned distributions can be approximated by mixtures of spread components, enabling the stretched-exponential decay in the general GHZ-support setting (beyond XOR structure). These results significantly advance understanding of multiplayer parallel repetition, with implications for complexity theory and high-dimensional combinatorics, by moving from polynomial to stretched-exponential decay under a natural, broad query-support condition. The methods also yield a concentration bound for the GHZ game, strengthening prior exponential decay results and offering new tools for analyzing high-dimensional combinatorial games through algebraic pseudorandomness.

Abstract

We prove that for any 3-player game , whose query distribution has the same support as the GHZ game (i.e., all satisfying ), the value of the -fold parallel repetition of decays exponentially fast: for all sufficiently large , where is an absolute constant. We also prove a concentration bound for the parallel repetition of the GHZ game: For any constant , the probability that the players win at least a fraction of the coordinates is at most , where 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 . Our key technical tool is the notion of \emph{algebraic spreadness} adapted from the breakthrough work of Kelley and Meka (FOCS '23) on sets free of 3-term progressions.
Paper Structure (32 sections, 29 theorems, 98 equations)

This paper contains 32 sections, 29 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Overview
  4. General Inductive Framework and High-Level Approach
  5. Step 1: Squares are Hard
  6. Step 2: Approximating the Distribution
  7. Uniformization:
  8. Counting and Approximation:
  9. Why Algebraic Spreadness?
  10. Preliminaries
  11. Probability Distributions
  12. Vector Spaces over GF(2)
  13. 3-Player Games and Parallel Repetition
  14. Some Basic Results on Parallel Repetition
  15. Algebraic Spreadness and Uniformization
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $\mathcal{G}$ be any 3-player game with value $\textnormal{val}(\mathcal{G})<1$, whose query distribution has support Then, for all sufficiently large $n$,i.e., $n\geq N$, where $N$ is a constant depending on the game $\mathcal{G}$. it holds that where $c>0$ is an absolute constant.

Theorems & Definitions (65)

  • Theorem 1.1: Parallel Repetition for the GHZ query support
  • Theorem 1.2: A Concentration Bound; restated and proved as Theorem \ref{['thm:ghz_conc']}
  • Corollary 1.3
  • Definition 2.1: Diagonal-Product Set
  • Definition 2.2: Square
  • Remark 2.3
  • Lemma 3.2
  • proof
  • Definition 3.3: Inner Product
  • Definition 3.4: $L^p$ norm
  • ...and 55 more