We investigate quantum many-body systems from the perspective of quantum computation, with a focus on discrete and continuous geometrical features emerging in such systems. In this context, geometry appears in many closely related settings: Tensor network models endow quantum systems with effective geometries determined by their patterns of many-body entanglement; quantum error-correcting codes protect logical quantum information through the geometry of their encoding into physical degrees of freedom; and holographic bulk/boundary dualities suggest that the geometry of spacetime itself can emerge from microscopic quantum systems. We develop theoretical methods that identify common structures across these areas, drawing motivation both from fundamental questions in high-energy theoretical physics and from applied problems in quantum computation and quantum information processing.