We study the complexity of computing the commuting-operator value ω∗
of entangled XOR games with any number of players. We introduce necessary and
sufficient criteria for an XOR game to have ω∗=1, and use these
criteria to derive the following results:
1. An algorithm for symmetric games that decides in polynomial time whether
ω∗=1 or ω∗<1, a task that was not previously known to be
decidable, together with a simple tensor-product strategy that achieves value 1
in the former case. The only previous candidate algorithm for this problem was
the Navascu\'{e}s-Pironio-Ac\'{i}n (also known as noncommutative Sum of Squares
or ncSoS) hierarchy, but no convergence bounds were known.
2. A family of games with three players and with ω∗<1, where it
takes doubly exponential time for the ncSoS algorithm to witness this (in
contrast with our algorithm which runs in polynomial time).
3. A family of games achieving a bias difference 2(ω∗−ω)
arbitrarily close to the maximum possible value of 1 (and as a consequence,
achieving an unbounded bias ratio), answering an open question of Bri\"{e}t and
Vidick.
4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games:
that is, we show that there exists a constant Ckunsat depending
only on the number k of players, such that a random k-XOR game over an
alphabet of size n has ω∗<1 with high probability when the number
of clauses is above Ckunsatn.
5. A lower bound of Ω(nlog(n)/loglog(n)) on the number of levels
in the ncSoS hierarchy required to detect unsatisfiability for most random
3-XOR games. This is in contrast with the classical case where the n-th level
of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all
possible solutions.Comment: 55 page