  The geometry and topology of quantum systems have deep connections to quantum
dynamics. In this paper, I show how to measure the non-Abelian Berry curvature
and its related topological invariant, the second Chern number, using dynamical
techniques. The second Chern number is the defining topological characteristic
of the four-dimensional generalization of the quantum Hall effect and has
relevance in systems from three-dimensional topological insulators to
Yang-Mills field theory. I illustrate its measurement using the simple example
of a spin-3/2 particle in an electric quadrupole field. I show how one can
dynamically measure diagonal components of the Berry curvature in an
over-complete basis of the degenerate ground state space and use this to
extract the full non-Abelian Berry curvature. I also show that one can
accomplish the same ideas by stochastically averaging over random initial
states in the degenerate ground state manifold. Finally I show how this system
can be manufactured and the topological invariant measured in a variety of
realistic systems, from superconducting qubits to trapped ions and cold atoms.
