The two pillars on which modern physics rests, quantum theory and general relativity, remain to be reconciled. One of the key problems is understanding the notion of causal structure -- the network of cause-effect relations that hold among events. On the one hand, causal structure provides the scaffolding underlying the fabric of space and time and on the other hand it struggles, in its conventional form, to accommodate the most mysterious (and exciting) features of quantum theory. This project aims to determine what innovations to the concept of causal structure are implied by quantum theory, to uncover the implications for the foundations of physics, and to leverage the quantum novelty as a resource for practical applications. Questions we aim to answer include: Can there be quantum superpositions of different causal structures and if so how might they be observed experimentally? If causal structure is indefinite, how does one define salient features of space and time, such as distance, curvature, and local time? What do quantum innovations to the notion of causal structure imply for the problem of deducing causal relations from observed correlations? What do they imply about the ultimate limits to information processing? The consortium we propose encompasses a broad diversity of approaches to the problem and includes the inventors of many of the techniques that will be critical to its solution. Through regular meetings and visits between the nodes, we aim to pool our varied perspectives and intellectual resources. The results will be disseminated through scholarly publications, two workshops, a conference, a summer school, and several public lectures. We expect our project to intensify research on the subject of quantum causal structures, foster greater collaboration between researchers studying gravitational physics, the foundations of quantum theory, quantum information theory, and machine learning, and to inspire new experiments and new quantum technologies.