  Matrix-product states and their continuous analogues are variational classes
of states that capture quantum many-body systems or quantum fields with low
entanglement; they are at the basis of the density-matrix renormalization group
method and continuous variants thereof. In this work we show that, generically,
N-point functions of arbitrary operators in discrete and continuous
translationally invariant matrix product states are completely characterized by
the corresponding two- and three-point functions. Aside from having important
consequences for the structure of correlations in quantum states with low
entanglement, this result provides a new way of reconstructing unknown states
from correlation measurements, e.g., for one-dimensional continuous systems of
cold atoms. We argue that such a relation of correlation functions may help in
devising perturbative approaches to interacting theories.
