  We introduce classical and quantum no-signalling bicorrelations and
characterise the different types thereof in terms of states on operator system
tensor products, exhibiting connections with bistochastic operator matrices and
with dilations of quantum magic squares. We define concurrent bicorrelations as
a quantum input-output generalisation of bisynchronous correlations. We show
that concurrent bicorrelations of quantum commuting type correspond to tracial
states on the universal C*-algebra of the projective free unitary quantum
group, showing that in the quantum input-output setup, quantum permutations of
finite sets must be replaced by quantum automorphisms of matrix algebras. We
apply our results to study the quantum graph isomorphism game, describing the
game C*-algebra in this case, and make precise connections with the algebraic
notions of quantum graph isomorphism, existing presently in the literature.
