  We show via explicit construction that all the extreme rays of both the
three-party quantum entropy cone and the four-party stabilizer entropy cone can
be obtained from subsystem coarse grainings of specific higher-party quantum
states, namely extreme states characterized by saturating a (non-trivial)
maximal set of instances of subadditivity. This suggests that the study of the
``subadditivity cone'', and the set of its extreme rays realizable in quantum
mechanics, provides a powerful tool for deriving inner bounds for the quantum
and stabilizer entropy cones, as well as constraints on new inequalities for
the von Neumann entropy.
