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

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.