We explore connections between macroscopic wormholes, quantum entanglement, quantum circuit complexity, and quantum chaos in the context of the anti-de Sitter / conformal field theory (AdS/CFT) correspondence. We first describe the construction of a single-boundary wormhole geometry in type IIB supergravity, which has a dual interpretation in terms of entanglement between sectors of a Higgsed Super Yang-Mills theory. The construction involves a CPT twist in the gluing of the wormhole to the exterior throats that gives a global monodromy to some coordinates while preserving orientability. We argue that the solution can be made long-lived by appropriate choice of parameters, and comment on mechanisms for generating traversability. We also describe a construction of a double wormhole between two universes. Motivated by the study of entanglement structures that support wormhole topologies, we study how the quantum circuit complexity in field theory quantifies the difficulty of distributing entanglement among multiple parties. We develop an Euler-Arnold formalism for computing complexity based on Nielsen's geometrization of gate counting. Applying this formalism to Gaussian states of the harmonic oscillator that possess a multiparty entanglement structure analogous to multiboundary wormhole configurations, we find a scaling with entropy that resembles a result for the interior volume of holographic multiboundary wormholes. We also study the complexity of time evolution, which was recently conjectured to be related to properties of wormhole interiors. This complexity grows linearly at early times until the minimal geodesic on the unitary manifold encounters an obstruction in the form of a conjugate point or geodesic loop. By explicitly locating these obstructions through analytical and numerical techniques, we demonstrate a complexity hierarchy: the complexity of time evolution is upper bounded by $\mathcal{O}(\sqrt{N})$ in free theories, $\mathcal{O}(\text{poly}(N))$ in interacting integrable theories, and $\mathcal{O}(e^N)$ in chaotic theories. We discuss the interpretation of these results in AdS/CFT, where wormhole solutions demonstrate quantum chaotic behavior.