  Middle cohomology states on the Higgs branch of supersymmetric quiver quantum
mechanics - also known as pure Higgs states - have recently emerged as possible
microscopic candidates for single-centered black hole micro-states, as they
carry zero angular momentum and appear to be robust under wall-crossing. Using
the connection between quiver quantum mechanics on the Coulomb branch and the
quantum mechanics of multi-centered black holes, we propose a general algorithm
for reconstructing the full moduli-dependent cohomology of the moduli space of
an arbitrary quiver, in terms of the BPS invariants of the pure Higgs states.
We analyze many examples of quivers with loops, including all cyclic Abelian
quivers and several examples with two loops or non-Abelian gauge groups, and
provide supporting evidence for this proposal. We also develop methods to count
pure Higgs states directly.
