Membership in Moment Polytopes is in NP and coNP

We show that the problem of deciding membership in the moment polytope associated with a finite-dimensional unitary representation of a compact, connected Lie group is in NP and coNP. This is the first nontrivial result on the computational complexity of this problem, which naively amounts to a quadratically constrained program. Our result applies in particular to the Kronecker polytopes, and therefore to the problem of deciding positivity of the stretched Kronecker coefficients. In contrast, it has recently been shown that deciding positivity of a single Kronecker coefficient is NP-hard, in general [C. Ikenmeyer, K. D. Mulmuley, and M. Walter, preprint, arXiv:1507.02955, 2015]. We discuss the consequences of our work in the context of complexity theory and the quantum marginal problem.