Chaotic behaviour is ubiquitous and plays an important part in most fields of science. In classical physics, chaos is characterized by hypersensitivity of the time evolution of a system to initial conditions. Quantum mechanics does not permit a similar definition owing in part to the uncertainty principle, and in part to the Schrödinger equation, which preserves the overlap between quantum states. This fundamental disconnect poses a challenge to quantum–classical correspondence1, and has motivated a long-standing search for quantum signatures of classical chaos2, 3. Here we present the experimental realization of a common paradigm for quantum chaos—the quantum kicked top2, 4— and the observation directly in quantum phase space of dynamics that have a chaotic classical counterpart. Our system is based on the combined electronic and nuclear spin of a single atom and is therefore deep in the quantum regime; nevertheless, we find good correspondence between the quantum dynamics and classical phase space structures. Because chaos is inherently a dynamical phenomenon, special significance attaches to dynamical signatures such as sensitivity to perturbation1, 5 or the generation of entropy6 and entanglement7, 8, for which only indirect evidence has been available9, 10, 11. We observe clear differences in the sensitivity to perturbation in chaotic versus regular, non-chaotic regimes, and present experimental evidence for dynamical entanglement as a signature of chaos.