  Hypergraph states form a family of multiparticle quantum states that
generalizes cluster states and graph states. We study the action and graphical
representation of nonlocal unitary transformations between hypergraph states.
This leads to a generalization of local complementation and graphical rules for
various gates, such as the CNOT gate and the Toffoli gate. As an application,
we show that already for five qubits local Pauli operations are not sufficient
to check local equivalence of hypergraph states. Furthermore, we use our rules
to construct entanglement witnesses for three-uniform hypergraph states.
