An archimedean Positivstellensatz in the spirit of Putinar and Helton-McCullough is presented leading to a hierarchy of semidefinite relaxations converging monotonically to the optimum of a state polynomial. This hierarchy can be seen as an analog of the Lasserre hierarchy for optimization of polynomials, and the Navascués-Pironio-Acín scheme for optimization of noncommutative polynomials. The motivation behind this theory arises from the study of correlations in quantum networks. 

Determining the maximal quantum violation of a polynomial Bell inequality for an arbitrary network is reformulated as a state polynomial optimization problem. Several examples of quadratic Bell inequalities in the bipartite scenario are analyzed after exploiting sparsity and symmetry. This is based on a collaboration with Igor Klep, Jurij Volčič, and Jie Wang from the papers https://arxiv.org/abs/2301.12513 & https://arxiv.org/abs/2306.05761