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The quantum marginal problem consists in deciding whether a given set ofmarginal reductions is compatible with the existence of a global quantum stateor not. In this work, we formulate the problem from the perspective ofdynamical systems theory and study its advantages with respect to the standardapproach. The introduced formalism allows us to analytically determine globalquantum states from a wide class of self-consistent marginal reductions in anymultipartite scenario. In particular, we show that any self-consistent set ofmultipartite marginal reductions is compatible with the existence of a globalquantum state, after passing through a depolarizing channel. This resultreveals that the complexity associated to the marginal problem can bedrastically reduced when restricting the attention to sufficiently mixedmarginals. We also formulate the marginal problem in a compressed way, in thesense that the total number of scalar constraints is smaller than the onerequired by the standard approach. This fact suggests an exponential speedup inruntime when considering semi-definite programming techniques to solve it, inboth classical and quantum algorithms. Finally, we reconstruct $n$-qubitquantum states from all the $\binom{n}{k}$ marginal reductions to $k$ parties,generated from randomly chosen mixed states. Numerical simulations reveal thatthe fraction of cases where we can find a global state equals 1 when $5\leqn\leq12$ and $\lfloor(n-1)/\sqrt{2}\rfloor\leq k\leq n-1$, where$\lfloor\cdot\rfloor$ denotes the floor function.

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