  We analyze the structure of the space of temporal correlations generated by
quantum systems. We show that the temporal correlation space under dimension
constraints can be nonconvex. For the general case, we provide the necessary
and sufficient dimension of a quantum system needed to generate a convex
correlation space for a given scenario. We further prove that this dimension
coincides with the dimension necessary to generate any point in the temporal
correlation polytope. As an application of our results, we derive nonlinear
inequalities to witness the nonconvexity for qubits and qutrits in the simplest
scenario, and present an algorithm which can help to find the minimum for a
certain type of nonlinear expressions under dimension constraints.
