| Summary: | Causality is the notion that cause should precede effect. While its status as a fundamental physical principle is an issue of ongoing debate, its importance in the formulation of physical theories is not. In particular, the study of quantum theory usually assumes the existence of a fixed background structure. Under such an assumption, the causal order — e.g. temporal order — of events are fixed. If however, we allow the background structure to be subject to quantum uncertainty, we can obtain situations in which causal order is itself subject to quantum uncertainty — this is known as indefinite causal order. In this thesis, we used a recently-developed quantum mechanical framework known as the Process Matrix formalism in our description of such situations. We then used this framework in a search for physically-feasible resources that not only generate indefinite causal order, but also violate causal inequalities — upper limits of correlations produced by resources that generate only definite causal order. Violation of these inequalities are of interest as they might signify advantages in various communication tasks. While the search was unsuccessful, we managed to prove statements that exclude a larger class of resources from violating causal inequalities, thus further narrowing the search for these resources.
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