  We study the weak recovery problem on the $r$-uniform hypergraph stochastic
block model ($r$-HSBM) with two balanced communities. In this model, $n$
vertices are randomly divided into two communities, and size-$r$ hyperedges are
added randomly depending on whether all vertices in the hyperedge are in the
same community. The goal of weak recovery is to recover a non-trivial fraction
of the communities given the hypergraph. Pal and Zhu (2021); Stephan and Zhu
(2022) established that weak recovery is always possible above a natural
threshold called the Kesten-Stigum (KS) threshold. For assortative models
(i.e., monochromatic hyperedges are preferred), Gu and Polyanskiy (2023) proved
that the KS threshold is tight if $r\le 4$ or the expected degree $d$ is small.
For other cases, the tightness of the KS threshold remained open.
  In this paper we determine the tightness of the KS threshold for a wide range
of parameters. We prove that for $r\le 6$ and $d$ large enough, the KS
threshold is tight. This shows that there is no information-computation gap in
this regime and partially confirms a conjecture of Angelini et al. (2015). On
the other hand, we show that for $r\ge 5$, there exist parameters for which the
KS threshold is not tight. In particular, for $r\ge 7$, the KS threshold is not
tight if the model is disassortative (i.e., polychromatic hyperedges are
preferred) or $d$ is large enough. This provides more evidence supporting the
existence of an information-computation gap in these cases.
  Furthermore, we establish asymptotic bounds on the weak recovery threshold
for fixed $r$ and large $d$. We also obtain a number of results regarding the
broadcasting on hypertrees (BOHT) model, including the asymptotics of the
reconstruction threshold for $r\ge 7$ and impossibility of robust
reconstruction at criticality.
