The H-index of a node in a static network is the maximum value $h$ such that at least $h$ of its neighbors have a degree of at least $h$. Recently, a generalized version, the $n$-th order H-index, was introduced, allowing to relate degree centrality, H-index, and the $k$-core of a node. We extend the $n$-th order H-index to temporal networks and define corresponding temporal centrality measures and temporal core decompositions. Our $n$-th order temporal H-index respects the reachability in temporal networks leading to node rankings, which reflect the importance of nodes in spreading processes. We derive natural decompositions of temporal networks into subgraphs with strong temporal coherence. We analyze a recursive computation scheme and develop a highly scalable streaming algorithm. Our experimental evaluation demonstrates the efficiency of our algorithms and the conceptional validity of our approach. Specifically, we show that the $n$-th order temporal H-index is a strong heuristic for identifying super-spreaders in evolving social networks and detects temporally well-connected components.