  We study new cohomologies for the BPS operators of the $\mathcal{N}=4$
Yang-Mills theory with $SU(3)$ and $SU(4)$ gauge groups, to better understand
the black hole microstates. We first study the index of these black hole
operators and identify their apparent threshold levels. For $SU(3)$, we find
many towers of states and partial no-hair behaviors. We explicitly construct
the threshold cohomology in the $SU(3)$ theory. We study throughout this paper
a subsector of the field theory corresponding to the BMN matrix theory. We also
argue that the BMN sector exhibits a black hole like entropy growth at large
$N$.
