  We introduce "binding complexity", a new notion of circuit complexity which
quantifies the difficulty of distributing entanglement among multiple parties,
each consisting of many local degrees of freedom. We define binding complexity
of a given state as the minimal number of quantum gates that must act between
parties to prepare it. To illustrate the new notion we compute it in a toy
model for a scalar field theory, using certain multiparty entangled states
which are analogous to configurations that are known in AdS/CFT to correspond
to multiboundary wormholes. Pursuing this analogy, we show that our states can
be prepared by the Euclidean path integral in $(0+1)$-dimensional quantum
mechanics on graphs with wormhole-like structure. We compute the binding
complexity of our states by adapting the Euler-Arnold approach to Nielsen's
geometrization of gate counting, and find a scaling with entropy that resembles
a result for the interior volume of holographic multiboundary wormholes. We
also compute the binding complexity of general coherent states in perturbation
theory, and show that for "double-trace deformations" of the Hamiltonian the
effects resemble expansion of a wormhole interior in holographic theories.
