We study the entanglement properties of quantum hypergraph states of $n$
qubits, focusing on multipartite entanglement. We compute multipartite
entanglement for hypergraph states with a single hyperedge of maximum
cardinality, for hypergraph states endowed with all possible hyperedges of
cardinality equal to $n-1$ and for those hypergraph states with all possible
hyperedges of cardinality greater than or equal to $n-1$. We then find a lower
bound to the multipartite entanglement of a generic quantum hypergraph state.
We finally apply the multipartite entanglement results to the construction of
entanglement witness operators, able to detect genuine multipartite
entanglement in the neighbourhood of a given hypergraph state. We first build
entanglement witnesses of the projective type, then propose a class of
witnesses based on the stabilizer formalism, hence called stabilizer witnesses,
able to reduce the experimental effort from an exponential to a linear growth
in the number of local measurement settings with the number of qubits.