In this talk, I will explain how to understand the symplectic structure of the gravitational phase space in AdS in terms of the quantum overlap in the holographically dual CFT. As an application, I will use this to study the boundary description of the volume of maximal Cauchy slices. I will construct the boundary deformation that is conjugate to the volume in empty AdS, and the thermofield double at infinite time. I will explain the bulk interpretation of this deformation and speculate on its boundary interpretation. This will motivate a concrete version of the complexity equals volume conjecture, where the boundary complexity is defined as the energy of geodesics in a Kahler geometry of half sided Euclidean sources. I will calculate this complexity for states dual to Banados geometries that are close to the vacuum, and discuss a mini superspace approximation in the case of the time dependent thermofield double.