  The correlations arising from sequential measurements on a single quantum
system form a polytope. This is defined by the arrow-of-time (AoT) constraints,
meaning that future choices of measurement settings cannot influence past
outcomes. We discuss the resources needed to simulate the extreme points of the
AoT polytope, where resources are quantified in terms of the minimal dimension,
or "internal memory" of the physical system. First, we analyze the equivalence
classes of the extreme points under symmetries. Second, we characterize the
minimal dimension necessary to obtain a given extreme point of the AoT
polytope, including a lower scaling bound in the asymptotic limit of long
sequences. Finally, we present a general method to derive dimension-sensitive
temporal inequalities for longer sequences, based on inequalities for shorter
ones, and investigate their robustness to imperfections.
