A path cover of a digraph is a collection of paths collectively containing the vertex set of the digraph. A path cover with minimum cardinality for a directed acyclic graph can be found in polynomial time [Fulkerson, AMS ’56; Cáceres et al., SODA’22]. Moreover, Dilworth’s celebrated theorem on chain coverings of partially ordered sets equivalently states that the minimum size of a path cover of a DAG is equal to the maximum size of a set of mutually unreachable vertices. In this paper, we examine how far Dilworth’s theorem can be extended to a “dynamic” analogue of directed acyclic graphs. A temporal digraph has an arc set that changes over discrete time-steps. Furthermore, if the underlying digraph (i.e., the union of all the arc sets that appears at some point) is acyclic, then we have a temporal directed acyclic graph (or simply a temporal DAG). A temporal path is a directed path in the underlying digraph, such that the time-steps of arcs are strictly increasing along the path. Two temporal paths are temporally disjoint if they do not occupy any vertex at the same time. A temporal path cover is a collection C of temporal paths that covers all vertices. Furthermore, C is a temporally disjoint path cover if all temporal paths are pairwise temporally disjoint. In this paper, we study the computational complexities of the problems of finding a temporal (disjoint) path cover with minimum cardinality (denoted as Temporal Path Cover and Temporally Disjoint Path Cover). We show that both Temporal Path Cover and Temporally Disjoint Path Cover are NP-hard even when the underlying DAG is planar, bipartite, subcubic, and there are only two arc-disjoint time-steps. Moreover, Temporally Disjoint Path Cover remains NP-hard even on temporal oriented trees. We also observe that natural temporal analogues of Dilworth’s theorem on these classes of temporal DAGs do not hold. In contrast, we show that Temporal Path Cover is polynomial-time solvable on temporal oriented trees by a reduction to Clique Cover for (static undirected) weakly chordal graphs (a subclass of perfect graphs for which Clique Cover admits an efficient algorithm). This highlights an interesting algorithmic difference between the two problems. Although it is NP-hard on temporal oriented trees, Temporally Disjoint Path Cover becomes polynomial-time solvable on temporal oriented lines and temporal rooted directed trees. For all these positive algorithmic results, we also show that temporal analogues of Dilworth’s theorem hold for the corresponding temporal graph classes. We also show that Temporal Path Cover and Temporally Disjoint Path Cover become efficiently solvable when the number of time-steps is bounded and the underlying graph is close to a tree. More precisely, we show that Temporal Path Cover admits an XP time algorithm with respect to parameter tmax + tw, where tmax is the maximum time-step, and tw is the treewidth of the underlying static undirected graph. We also show that Temporally Disjoint Path Cover admits an FPT algorithm with respect to the same parameter.