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