  We introduce a sum-of-squares SDP hierarchy approximating the ground-state
energy from below for quantum many-body problems, with a natural quantum
embedding interpretation. We establish the connections between our approach and
other variational methods for lower bounds, including the variational
embedding, the RDM method in quantum chemistry, and the Anderson bounds.
Additionally, inspired by the quantum information theory, we propose efficient
strategies for optimizing cluster selection to tighten SDP relaxations while
staying within a computational budget. Numerical experiments are presented to
demonstrate the effectiveness of our strategy. As a byproduct of our
investigation, we find that quantum entanglement has the potential to capture
the underlying graph of the many-body Hamiltonian.
