In this talk, I will discuss correlations that can be generated by performing local measurements on bipartite quantum systems. I'll present an algebraic characterization of the set of quantum correlations which allows us to identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a quantum correlation. I will then discuss some examples showing the tightness of our lower bound. Also, the algebraic characterization can be used to express the set of quantum correlations as the projection of an affine section of the cone of completely positive semidefinite matrices. Using this, we identify a semidefinite programming outer approximation to the set of quantum correlations which is contained in the first level of the Navascués, Pironio and Acín hierarchy, and a linear conic programming problem formulating exactly the quantum value of a nonlocal game. Time permitting, I will discuss other consequences of these conic formulations and some interesting special cases. 

This talk is based on work with Antonios Varvitsiotis and Zhaohui Wei, arXiv:1507.00213 and arXiv:1506.07297.