  We show that for any eps>0 the problem of finding a factor (2-eps)
approximation to the entangled value of a three-player XOR game is NP-hard.
Equivalently, the problem of approximating the largest possible quantum
violation of a tripartite Bell correlation inequality to within any
multiplicative constant is NP-hard. These results are the first constant-factor
hardness of approximation results for entangled games or quantum violations of
Bell inequalities shown under the sole assumption that P \neq NP. They can be
thought of as an extension of Hastad's optimal hardness of approximation
results for MAX-E3-LIN2 (JACM'01) to the entangled-player setting.
  The key technical component of our work is a soundness analysis of a
point-vs-plane low-degree test against entangled players. This extends and
simplifies the analysis of the multilinearity test by Ito and Vidick (FOCS'12).
Our results demonstrate the possibility for efficient reductions between
entangled-player games and our techniques may lead to further hardness of
approximation results.
