We study the problem of finding the longest common sub-pattern (LCSP) shared by two sequences of temporal intervals. In particular we are interested in finding the LCSP of the corresponding arrangements. Arrangements of temporal intervals are a powerful way to encode multiple concurrent labeled events that have a time duration. Discovering commonalities among such arrangements is useful for a wide range of scientific fields and applications, as it can be seen by the number and diversity of the datasets we use in our experiments. In this paper, we define the problem of LCSP and prove that it is NP-complete by demonstrating a connection between graphs and arrangements of temporal intervals. This connection leads to a series of interesting open problems. In addition, we provide an exact algorithm to solve the LCSP problem, and also propose and experiment with three polynomial time and space under-approximation techniques. Finally, we introduce two upper bounds for LCSP and study their suitability for speeding up 1-NN search. Experiments are performed on seven datasets taken from a wide range of real application domains, plus two synthetic datasets. Lastly, we describe several application cases that demonstrate the need and suitability of LCSP.