  The spectral variant of the quantum marginal problem asks: Given prescribed
spectra for a set of quantum marginals, does there exist a compatible joint
state? The main idea of this work is a symmetry-reduced semidefinite
programming hierarchy for detecting incompatible spectra. The hierarchy is
complete, in the sense that it detects every incompatible set of spectra. The
refutations it provides are dimension-free, certifying incompatibility in all
local dimensions. The hierarchy equally applies to the sums of Hermitian
matrices problem, to optimize trace polynomials on the positive cone, to the
compatibility of invariants, and to certify vanishing Kronecker coefficients.
