Approximating the quantum value of an LCS game is RE-hard

Abstract

We generalize Håstad's long-code test for projection games and show that it remains complete and sound against entangled provers. Combined with a result of Dong et al. \cite{Dong25}, which establishes that $\MIP^*=\RE$ with constant-length answers, we derive that $\LIN^*_{1-ε,s}=\RE$, for some $1/2< s<1$ and for every sufficiently small $ε>0$, where LIN refers to linearity (over $\mathbb{F}_2$) of the verifier predicate. Achieving the same result with $ε=0$ would imply the existence of a non-hyperlinear group.

Theorems & Definitions (42)

• Definition 2.1: Fourier Transform
• Lemma 2.1: Fourier inversion formula and Parseval's identity
• proof
• Definition 2.2: Folding over true
• Lemma 2.2
• proof
• Definition 2.3: Conditioning upon a function
• Lemma 2.3
• proof
• Lemma 2.4