Credal networks are graph-based statistical models whose parameters take values in a set, instead of being sharply specified as in traditional statistical models (e.g.,~Bayesian networks). The computational complexity of inferences on such models depends on the irrelevance/independence concept adopted. In this paper, we study inferential complexity under the concepts of epistemic irrelevance and strong independence. We show that inferences under strong independence are NP-hard even in trees with ternary variables. We prove that under epistemic irrelevance the polynomial time complexity of inferences in credal trees is not likely to extend to more general models (e.g.~singly connected networks). These results clearly distinguish networks that admit efficient inferences and those where inferences are most likely hard, and settle several open questions regarding computational complexity.