  The class $\MIP^*$ of promise problems that can be decided through an
interactive proof system with multiple entangled provers provides a
complexity-theoretic framework for the exploration of the nonlocal properties
of entanglement. Little is known about the power of this class. The only
proposed approach for establishing upper bounds is based on a hierarchy of
semidefinite programs introduced independently by Pironio et al. and Doherty et
al. This hierarchy converges to a value that is only known to coincide with the
provers' maximum success probability in a given proof system under a plausible
but difficult mathematical conjecture, Connes' embedding conjecture. No bounds
on the rate of convergence are known.
  We introduce a rounding scheme for the hierarchy, establishing that any
solution to its $N$-th level can be mapped to a strategy for the provers in
which measurement operators associated with distinct provers have pairwise
commutator bounded by $O(\ell^2/\sqrt{N})$ in operator norm, where $\ell$ is
the number of possible answers per prover.
  Our rounding scheme motivates the introduction of a variant of $\MIP^*$,
called $\MIP_\delta^*$, in which the soundness property is required to hold as
long as the commutator of operations performed by distinct provers has norm at
most $\delta$. Our rounding scheme implies the upper bound $\MIP_\delta^*
\subseteq \DTIME(\exp(\exp(\poly)/\delta^2))$. In terms of lower bounds we
establish that $\MIP^*_{2^{-\poly}}$, with completeness $1$ and soundness
$1-2^{-\poly}$, contains $\NEXP$. The relationship of $\MIP_\delta^*$ to
$\MIPstar$ has connections with the mathematical literature on approximate
commutation. Our rounding scheme gives an elementary proof that the Strong
Kirchberg Conjecture implies that $\MIPstar$ is computable. We discuss
applications to device-independent cryptography.
