Recent advances in condensed matter, optical, and atomic physics led to the emergence of highly controllable synthetic quantum matter, such as superconducting circuits, implanted solid-state defects, trapped atoms or ions, and strongly interacting photons. In addition to allowing us to gain fundamental insights into peculiar and diverse behavior of many-body quantum systems, synthetic quantum matter paves the way for building revolutionary quantum technologies such as extraordinarily powerful computers, unbreakably secure communication devices, and exceptionally accurate sensors. In this talk, we will explore two facets of synthetic quantum matter. First, we will argue that sampling complexity, that is the question of how hard it is to produce a sample from a given probability distribution, lies at the heart of understanding and harnessing synthetic quantum matter. Second, we will show how to engineer interactions between individual photons and use these interactions for building quantum technologies and accessing exotic few-body and many-body physics.