Recently, there has been renewed interest in the physics of quantum non-integrable systems - systems that lack or have very few conservation laws - with implications discovered across diverse fields including quantum information, statistical mechanics, condensed matter and quantum gravity. For instance, highly excited states of non-integrable systems can store quantum information in a local noise-resistant manner, mimic equilibrium statistical ensembles, capture finite temperature phase transitions and harbor correlations that resemble those outside a blackhole embedded in hyperbolic space. Moreover, they have been predicted to exhibit various features in their temporal dynamics, including thermalization at long times and a quantum version of the "butterfly effect" that certifies classical chaos at short times. However, traditional algorithms for studying physics in this regime have exponential complexity, making access to it extremely challenging. We strive to develop techniques for studying non-integrable systems, unearthing their hidden equilibrium and dynamical properties and exploring their broader ramifications on other fields of physics.