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Moment polytopes characterize the asymptotic non-vanishing of representation-theoretic multiplicities, including the Kronecker coefficients of the symmetric groups. I will present a recent description that is computationally straightforward to use, and discuss some interesting implications - in particular that the corresponding membership problem is in NP and co-NP. In contrast, deciding positivity of an individual Kronecker coefficient is NP-hard in general. This suggests that representation theory can be easier in the asymptotic regime, and it also has implications on the complexity of the marginal problem in quantum physics. My talk is based on joint work with M. Vergne and with P. Buergisser, M. Christandl and K. Mulmuley.