For given quantum (non-commutative) spaces and , we study the quantum space of maps , from to . In case of finite quantum spaces, these objects turn out to be behind a large class of maps which generalize the classical qc-correlations known from quantum information theory to the setting of quantum input and output sets. We prove various operator algebraic properties of the C* -algebras C() such as the lifting property and residual finite dimensionality. Inside C() we construct a universal operator system related to and , and show, among other things, that the embedding C() is hyperrigid and has another interesting property, which we call the strong extension property. Furthermore, C() is the C* -envelope of and a large class of non-signalling correlations on the quantum sets and arise from states on C() ⊗ C() as well as states on the commuting tensor product ⊗. Finally, we introduce and study the notion of a synchronous correlation with quantum input and output sets and prove several characterizations of such correlations and their relation to traces on C()