  In this paper we study bipartite quantum correlations using techniques from
tracial noncommutative polynomial optimization. We construct a hierarchy of
semidefinite programming lower bounds on the minimal entanglement dimension of
a bipartite correlation. This hierarchy converges to a new parameter: the
minimal average entanglement dimension, which measures the amount of
entanglement needed to reproduce a quantum correlation when access to shared
randomness is free. For synchronous correlations, we show a correspondence
between the minimal entanglement dimension and the completely positive
semidefinite rank of an associated matrix. We then study optimization over the
set of synchronous correlations by investigating quantum graph parameters. We
unify existing bounds on the quantum chromatic number and the quantum stability
number by placing them in the framework of tracial optimization. In particular,
we show that the projective packing number, the projective rank, and the
tracial rank arise naturally when considering tracial analogues of the Lasserre
hierarchy for the stability and chromatic number of a graph. We also introduce
semidefinite programming hierarchies converging to the commuting quantum
chromatic number and commuting quantum stability number.
