Watch a 26-minute conference talk from TQC 2023 (Theory of Quantum Computation, Communication and Cryptography Conference) exploring a novel approach to the spectral quantum marginal problem through symmetry-reduced semidefinite programming. Learn how to detect incompatible spectra using a complete hierarchy that provides dimension-free refutations, applicable across multiple quantum computing domains. Delve into the mathematical framework that addresses the compatibility of quantum marginals, optimization of trace polynomials on positive cones, compatibility of invariants, and certification of vanishing Kronecker coefficients. Follow along as the speaker presents entropic inequalities, explores the swap trick, discusses relaxation for compatibility, and examines numerical results through various mathematical representations and permutations.