  Quantum Markov networks are a generalization of quantum Markov chains to
arbitrary graphs. They provide a powerful classification of correlations in
quantum many-body systems---complementing the area law at finite
temperature---and are therefore useful to understand the powers and limitations
of certain classes of simulation algorithms. Here, we extend the
characterization of quantum Markov networks and in particular prove the
equivalence of positive quantum Markov networks and Gibbs states of
Hamiltonians that are the sum of local commuting terms on graphs containing no
triangles. For more general graphs we demonstrate the equivalence between
quantum Markov networks and Gibbs states of a class of Hamiltonians of
intermediate complexity between commuting and general local Hamiltonians.
