  We introduce and examine three subclasses of the family of quantum
no-signalling (QNS) correlations introduced by Duan and Winter: quantum
commuting, quantum and local. We formalise the notion of a universal TRO of a
block operator isometry, define an operator system, universal for stochastic
operator matrices, and realise it as a quotient of a matrix algebra. We
describe the classes of QNS correlations in terms of states on the tensor
products of two copies of the universal operator system, and specialise the
correlation classes and their representations to classical-to-quantum
correlations. We study various quantum versions of synchronous no-signalling
correlations and show that they possess invariance properties for suitable sets
of states. We introduce quantum non-local games as a generalisation of
non-local games. We define the operation of quantum game composition and show
that the perfect strategies belonging to a certain class are closed under
channel composition. We specialise to the case of graph colourings, where we
exhibit quantum versions of the orthogonal rank of a graph as the optimal
output dimension for which perfect classical-to-quantum strategies of the graph
colouring game exist, as well as to non-commutative graph homomorphisms, where
we identify quantum versions of non-commutative graph homomorphisms introduced
by Stahlke.
