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Topological phases of matter serve as one of the most striking examples of the richness and novelty of quantum systems with many degrees of freedom.  In contrast to conventional matter, they are characterized by both non-local properties and non-classical notions such as entanglement.  I will discuss two broad categories of topological phases, distinguished by whether or not they possess fractionalized “anyon” excitations that are neither bosons nor fermions.  I will demonstrate that entanglement not only provides an understanding of such phases but also enables the transmutation between these two categories of topological phases. 

What is the essence of quantum theory? In the present talk I want to approach this question from a particular operationalist perspective. I take advantage of a recent convergence between operational approaches to quantum theory and axiomatic approaches to quantum field theory. Removing anything special to particular physical models, including underlying notions of space and (crucially) time, what remains is what I shall call "abstract quantum theory". This embeds into a more general framework that also includes classical theory, with classical and quantum theory representing two extremes in a spectrum of possible theories. I shall present this within a "hierarchy of abstraction", where adding structure leads to more specialized frameworks. In particular, adding topological spacetime and locality leads to the recently proposed positive formalism and recovers quantum field theory. Further rigidifying time and adding causality recovers the standard formulation of quantum theory. There are other promising specializations such as Hardy's proposal to work in op-space.