This special physics seminar from the Institute for Advanced Study features Matthew Hastings from Microsoft discussing the classification of quantum cellular automata (QCA) and their relationship to invertible phases of matter. Explore the formal definition of QCA as *-automorphisms of the algebra of local operators, distinguishing between trivial examples like local quantum circuits and nontrivial QCA that cannot be expressed as local circuits. Learn about the classification of nontrivial QCA across different spatial dimensions, including the one-dimensional "shift" index classification, the triviality of two-dimensional QCA (up to shifts), and the construction of nontrivial QCA in three or more dimensions using discrete Chern-Simons functionals. Discover the conjectured correspondence between unoriented cobordism (which classifies symmetry protected topological phases) and families of QCA, based on joint work with L. Fidkowski and J. Haah.