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 ${\text{MIP}}^{\ast }=\text{RE}$ with constant-length answers, we derive that ${\text{LIN}}_{1-\epsilon ,s}^{\ast }=\text{RE}$, for some $1\mathrm{/}2<s<1$ 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.