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