  We develop a general framework to construct quantum algorithms that detect if
a $3$-uniform hypergraph given as input contains a sub-hypergraph isomorphic to
a prespecified constant-sized hypergraph. This framework is based on the
concept of nested quantum walks recently proposed by Jeffery, Kothari and
Magniez [SODA'13], and extends the methodology designed by Lee, Magniez and
Santha [SODA'13] for similar problems over graphs. As applications, we obtain a
quantum algorithm for finding a $4$-clique in a $3$-uniform hypergraph on $n$
vertices with query complexity $O(n^{1.883})$, and a quantum algorithm for
determining if a ternary operator over a set of size $n$ is associative with
query complexity $O(n^{2.113})$.
