  In this work we study the sets of two-party correlations generated from a
Bell scenario involving two spatially separated systems with respect to various
physical models. We show that the sets of classical, quantum, no-signaling and
unrestricted correlations can be expressed as projections of affine sections of
appropriate convex cones. As a by-product, we identify a spectrahedral outer
approximation to the set of quantum correlations which is contained in the
first level of the Navascu\'es, Pironio and Ac\'in (NPA) hierarchy and also a
sufficient condition for the set of quantum correlations to be closed.
Furthermore, by our conic formulations, the value of a nonlocal game over the
sets of classical, quantum, no-signaling and unrestricted correlations can be
cast as a linear conic program. This allows us to show that a semidefinite
programming upper bound to the classical value of a nonlocal game introduced by
Feige and Lov\'asz is in fact an upper bound to the quantum value of the game
and moreover, it is at least as strong as optimizing over the first level of
the NPA hierarchy. Lastly, we show that deciding the existence of a perfect
quantum (resp. classical) strategy is equivalent to deciding the feasibility of
a linear conic program over the cone of completely positive semidefinite
matrices (resp. completely positive matrices). By specializing the results to
synchronous nonlocal games, we recover the conic formulations for various
quantum and classical graph parameters that were recently derived in the
literature.
