  Quantum information leverages properties of quantum behaviors in order to
perform useful tasks such as secure communication and randomness certification.
Nevertheless, not much is known about the intricate geometric features of the
set quantum behaviors. In this paper we study the structure of the set of
quantum correlators using semidefinite programming. Our main results are (i) a
generalization of the analytic description by Tsirelson-Landau-Masanes, (ii)
necessary and sufficient conditions for extremality and exposedness, and (iii)
an operational interpretation of extremality in the case of two dichotomic
measurements, in terms of self-testing. We illustrate the usefulness of our
theoretical findings with many examples and extensive computational work.
