  The question whether a given quantum state is a ground or thermal state of a
few-body Hamiltonian can be used to characterize the complexity of the state
and is important for possible experimental implementations. We provide methods
to characterize the states generated by two- and, more generally, k-body
Hamiltonians as well as the convex hull of these sets. This leads to new
insights into the question which states are uniquely determined by their
marginals and to a generalization of the concept of entanglement. Finally,
certification methods for quantum simulation can be derived.
