Entanglement marginal problems

We consider the entanglement marginal problem, which consists of deciding whether a number reduced density matrices are compatible with an overall separable quantum state. To tackle this we propose hierarchies semidefinite programming relaxations set state marginals admitting fully extension. connect completeness each hierarchy to resolution analog classical problem and thus identify relevant experimental situations where complete. For finitely many parties on star configuration or chain, find that can achieve arbitrarily good approximation nearest-neighbour states time (space) complexity polynomial (linear) system size. Our results even extend infinite systems, such as translation-invariant systems in 1D, well higher spatial dimensions extra symmetries.