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.

翻译：我们推广了Håstad针对投影游戏的长码检验，并证明其在纠缠证明者场景下仍然保持完备性和可靠性的。结合Dong等人[Dong25]的结果（该结果确立了在常数长度回答条件下MIP* = RE），我们推导出对于某个1/2 < s < 1以及每个足够小的ε > 0，有LIN*_{1-ε,s} = RE，其中LIN表示验证器谓词的线性性（在F_2上）。若在ε=0时得到相同结论，则意味着非超线性群的存在性。