  Many problems of quantum information theory rely on the set of quantum
marginals. A precise knowledge of the faces of this convex set is necessary,
for example, in the reconstruction of states from their marginals or in the
evaluation of complexity measures of many-body systems. Yet, even the two-body
marginals of just three qubits were only described in part. Here, we propose an
experimental method to search for the coatoms in the lattice of exposed faces
of the convex set of quantum marginals. The method is based on sampling from
the extreme points of the dual spectrahedron. We provide an algebraic
certificate of correctness, employing ground projectors of local Hamiltonians.
Using this method, we present an explicit family of coatoms of rank five in the
lattice of ground projectors of two-local three-qubit Hamiltonians (the rank is
always six for bits). This family describes a family of coatoms in the lattice
of exposed faces of the convex set of two-body marginals of three qubits.
Besides introducing the experimental method, we show that the support sets of
probability distributions that factor are the ground projectors of
frustration-free Hamiltonians in the commutative setting. We also discuss
nonexposed points of the set of marginals.
