  We show that given an explicit description of a multiplayer game, with a
classical verifier and a constant number of players, it is QMA-hard, under
randomized reductions, to distinguish between the cases when the players have a
strategy using entanglement that succeeds with probability 1 in the game, or
when no such strategy succeeds with probability larger than 1/2. This proves
the "games quantum PCP conjecture" of Fitzsimons and the second author
(ITCS'15), albeit under randomized reductions. The core component in our
reduction is a construction of a family of two-player games for testing
$n$-qubit maximally entangled states. For any integer $n\geq2$, we give a test
in which questions from the verifier are $O(\log n)$ bits long, and answers are
$\mathrm{poly}(\log\log n)$ bits long. We show that for any constant
$\varepsilon\geq0$, any strategy that succeeds with probability at least
$1-\varepsilon$ in the test must use a state that is within distance
$O(\varepsilon^c)$ from a state that is locally equivalent to a maximally
entangled state on $n$ qubits, for some universal constant $c>0$. The
construction is based on the classical plane-vs-point test for multivariate
low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test
to the quantum regime by executing independent copies of the test in the
generalized Pauli $X$ and $Z$ bases over $\mathbb{F}_q$, where $q$ is a
sufficiently large prime power, and combine the two through a test for the
Pauli twisted commutation relations. Our main complexity-theoretic result is
obtained by combining this family of games with constructions of PCPs of
proximity introduced by Ben-Sasson et al. (CCC'05), and crucially relies on a
linear property of such PCPs. Another consequence of our results is a
deterministic reduction from the games quantum PCP conjecture to a suitable
formulation of the Hamiltonian quantum PCP conjecture.
