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 $\epsilon >0$, 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.