Our earlier paper "Complexity Equals Action" conjectured that the quantum computational complexity of a holographic state is given by the classical action of a region in the bulk (the 'Wheeler-DeWitt' patch). We provide calculations for the results quoted in that paper; explain how it fits into a broader (tensor) network of ideas; and elaborate on the hypothesis that black holes are fastest computers in nature.

We conjecture that the quantum complexity of a holographic state is dual to the action of a certain spacetime region that we call a Wheeler-DeWitt patch. We illustrate and test the conjecture in the context of neutral, charged, and rotating black holes in anti-de Sitter spacetime, as well as black holes perturbed with static shells and with shock waves. This conjecture evolved from a previous conjecture that complexity is dual to spatial volume, but appears to be a major improvement over the original. In light of our results, we discuss the hypothesis that black holes are the fastest computers in nature.

We give arguments for the existence of a thermodynamics of quantum complexity that includes a "Second Law of Complexity". To guide us, we derive a correspondence between the computational (circuit) complexity of a quantum system of K qubits, and the positional entropy of a related classical system with 2 K degrees of freedom. We also argue that the kinetic entropy of the classical system is equivalent to the Kolmogorov complexity of the quantum Hamiltonian. We observe that the expected pattern of growth of the complexity of the quantum system parallels the growth of entropy of the classical system. We argue that the property of having less-thanmaximal complexity (uncomplexity) is a resource that can be expended to perform directed quantum computation.Although this paper is not primarily about black holes, we find a surprising interpretation of the uncomplexity-resource as the accessible volume of spacetime behind a black hole horizon.

As time passes, once simple quantum states tend to become more complex. For strongly coupled k-local Hamiltonians, this growth of computational complexity has been conjectured to follow a distinctive and universal pattern. In this paper we show that the same pattern is exhibited by a much simpler system-classical geodesics on a compact two-dimensional geometry of uniform negative curvature. This striking parallel persists whether the system is allowed to evolve naturally or is perturbed from the outside.

A model of cosmological inflation is proposed in which field space is a hyperbolic plane. The inflaton never slow-rolls, and instead orbits the bottom of the potential, buoyed by a centrifugal force. Though initial velocities redshift away during inflation, in negatively curved spaces angular momentum naturally starts exponentially large and remains relevant throughout. Quantum fluctuations produce perturbations that are adiabatic and approximately scale invariant; strikingly, in a certain parameter regime the perturbations can grow double exponentially during horizon crossing.

The Jackiw-Teitelboim (JT) model arises from the dimensional reduction of charged black holes. Motivated by the holographic complexity conjecture, we calculate the late-time rate of change of action of a Wheeler-DeWitt patch in the JT theory. Surprisingly, the rate vanishes. This is puzzling because it contradicts both holographic expectations for the rate of complexification and also action calcula-

We derive the rate for transitions between de Sitter vacua by treating the field theory on the static patch as a thermal system. This reproduces the Coleman-De Luccia formalism for calculating the rate, but leads to a modified interpretation of the bounce solution and a different prediction for the evolution of the system after tunneling. The bounce is seen to correspond to a sequence of configurations interpolating between initial and final configurations on either side of the tunneling barrier, all of which are restricted to the static patch. The final configuration, which gives the initial data on the static patch for evolution after tunneling, is obtained from one half of a slice through the center of the bounce, while the other half gives the configuration before tunneling.The formalism makes no statement about the fields beyond the horizon.This approach resolves several puzzling aspects and interpretational issues concerning the Coleman-De Luccia and Hawking-Moss bounces. We work in the limit where the back reaction of matter on metric can be ignored, but argue that the qualitative aspects remain in the more general case. The extension to tunneling between anti-de Sitter vacua is discussed.

The growth of the "size" of operators is an important diagnostic of quantum chaos. In [1] it was conjectured that the holographic dual of the size is proportional to the average radial component of the momentum of the particle created by the operator. Thus the growth of operators in the background of a black hole corresponds to the acceleration of the particle as it falls toward the horizon.In this note we will use the momentum-size correspondence as a tool to study scrambling in the field of a near-extremal charged black hole. The agreement with previous work provides a non-trivial test of the momentum-size relation, as well as an explanation of a paradoxical feature of scrambling previously discovered by Leichenauer [2]. Naively Leichenauer's result says that only the non-extremal entropy participates in scrambling. The same feature is also present in the SYK model.In this paper we find a quite different interpretation of Leichenauer's result which does not have to do with any decoupling of the extremal degrees of freedom. Instead it has to do with the buildup of momentum as a particle accelerates through the long throat of the Reissner-Nordström geometry.Version 3: in this version of the paper, the conjectured size-momentum relation has been significantly modified. Rather than the proportionality factor being constant, we now conjecture that it varies through the throat. This result also agrees with forthcoming direct calculations in SYK.