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

TL;DR

This paper proves that approximating the quantum value of an LCS game is RE-hard, extending H {a}stad's long-code test and connecting it to the complexity class RE through entangled provers.

Contribution

It generalizes H {a}stad's test for projection games and links the complexity of quantum games to the class RE, showing RE-hardness for approximating quantum values.

Findings

Generalized H {a}stad's long-code test for projection games.

Established RE-hardness of approximating the quantum value of LCS games.

Connected the problem to the non-hyperlinear group conjecture.

Abstract

We generalize H\r{a}stad'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 $\LIN^*_{1-\epsilon,s}=\RE$, for some $1/2< s<1$ and for every sufficiently small $\epsilon>0$, where LIN refers to linearity (over $\mathbb{F}_2$) of the verifier predicate. Achieving the same result with $\epsilon=0$ would imply the existence of a non-hyperlinear group.