A fundamental area in statistical analysis is the study of which causal structures connecting the events of interest can explain the correlations that are observed between them. This is done through the falsification of invalid causal models. Our causal structure might posit the existence of hidden (unobserved) causes between the observed events. For example, if we see a positive correlation between the numbers of shark attacks and ice cream sales, we do not expect to explain it by a direct causal influence between these two things; instead, there should be a hidden common cause (for example, the Summer) that explains the correlation. Physicists also have a vested interest in falsifying causal hypotheses involving hidden variables. Bell's Theorem, for example, highlights the failure of many such classical causal hypotheses to explain the correlations predicted by quantum theory. In the scenario which Bell considered, if instead of treating the unobserved causes of classical random variables we treat them as potentially entangled quantum systems, we can explain a strictly larger set of correlations. Out project explores a simple but difficult question: In what other causal structures this also happens? In other words, for a given causal hypothesis, would the set of correlations it can explain expand if we relax our assumptions regarding posited unobservable systems to allow for shared entanglement? By a series of tricks developed during the PSI Winter School, we found that allowing for quantum causes makes an operational difference in a large number of causal hypotheses involving four observed variables. This work is of general interest as it generalizes Bell's Theorem: it exposes (qualitatively novel?!) advantages afforded by quantum theory over classical models. Bell's Theorem has proven crucially insightful in efforts to provide a causal accounting of quantum theory, and has inspired a plethora of quantum information theoretic protocols; similar dividends may be implicitly suggested by this work.