  Rapidly growing capabilities of quantum simulators to probe quantum many-body
phenomena require new methods to characterize increasingly complex states. We
present a protocol that constrains quantum states by experimentally measured
correlation functions which only scales polynomially with system size. This
method enables measurement of a quantum state's entanglement structure, opening
a new route to study entanglement-related phenomena. Our approach extends
Gaussian state parameterizations by systematically incorporating higher-order
correlations. We show the protocol's usefulness in conjunction with current and
forthcoming experimental capabilities, focusing on weakly interacting fermions
as a proof of concept. Here, the lowest non-trivial expansion quantitatively
predicts early time thermalization dynamics, including signaling the on-set of
quantum chaos indicated by the entanglement Hamiltonian.
