Exotic quantum systems -- those with macroscopic quantum behavior with no classical analogue -- have been a mainstay of condensed matter theory since the discovery of the quantum Hall effect. Since then, several families of related systems have been uncovered. Some, such as topological insulators, were predicted first and found experimentally later. Others are more elusive, like strongly correlated bosonic symmetry protected topological phases in high dimension, of which concrete evidence is still lacking. In this dissertation, we study several examples of exotic quantum systems. For SPT phases, we present a physically motivated classification scheme for interacting bosons and a bulk signature independent of boundary. We then construct a new, beyond Landau-Ginzburg second-order phase transition between two ordered phases of the Heisenberg magnet on the triangular lattice. Finally, we investigate an infinite family of spin liquid states, and conjecture on their connection to black holes.
In the vicinity of a phase transition ergodicity can be broken. Here, different initial many-body configurations evolve towards one of several fixed points, which are macroscopically distinguishable through an order parameter. This mechanism enables state classification in quantum cellular automata and feed-forward quantum neural networks. We demonstrate that this capability can be efficiently learned from training data even in extremely high-dimensional state spaces. We illustrate this using a quantum cellular automaton that allows binary classification, which is closely connected to the dynamics of a
Z2-symmetric Ising model with local interactions and dissipation. This approach can be generalized beyond binary classification and offers a natural framework for exploring the link between emergent many-body phenomena and the interpretation of data processing capabilities in the context of quantum machine learning.