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

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.