  We investigate the hypersymmetry bounds on the higher spin black hole
parameters that follow from the asymptotic symmetry superalgebra in higher-spin
anti-de Sitter gravity in three spacetime dimensions. We consider anti-de
Sitter hypergravity for which the analysis is most transparent. This is a
$osp(1\vert 4) \oplus osp(1\vert 4)$ Chern-Simons theory which contains,
besides a spin-$2$ field, a spin-$4$ field and a spin-$5/2$ field. The
asymptotic symmetry superalgebra is then the direct sum of two-copies of the
hypersymmetric extension $W_{(2,\frac52,4)}$ of $W_{(2,4)}$, which contains
fermionic generators of conformal weight $5/2$ and bosonic generators of
conformal weight $4$ in addition to the Virasoro generators. Following standard
methods, we derive bounds on the conserved charges from the anticommutator of
the hypersymmetry generators. The hypersymmetry bounds are nonlinear and are
saturated by the hypersymmetric black holes, which turn out to possess
$1/4$-hypersymmetry and to be "extreme", where extremality can be defined in
terms of the entropy: extreme black holes are those that fulfill the
extremality bounds beyond which the entropy ceases to be a real function of the
black hole parameters. We also extend the analysis to other $sp(4)$-solitonic
solutions which are maximally (hyper)symmetric.
