In this talk I will show that quantum many-body states satisfying an area law for entanglement have a local entanglement spectrum, i.e. the entanglement spectrum can be approximated by the spectrum of a local model acting on the boundary of the region. The result follows from a version of the Hammersley-Clifford Theorem (which states that classical Gibbs states are equivalent to Markov networks) for quantum approximate Markov chains. In particular I'll argue that those are in one-to-one correspondence to 1D quantum Gibbs states.