Graphs describe pairwise relations, while hypergraphs represent interactions involving more than two vertices. Super-HyperGraphs further permit vertices to be selected from iterated powersets, so that incidences can occur among nested objects such as teams, clusters, departments, portfolios, or control units. Many systems with such hierarchical organization are also time-dependent: their relations appear, disappear, or change activity over discrete or continuous time. Existing temporal graphs and temporal hypergraphs record temporal activation of edges or hyperedges, but they usually operate over a single-level vertex domain and therefore do not retain the identity of higher-level interacting objects. This paper develops the temporal \(n\)-Super-HyperGraph as a time-labeled higher-order structure for dynamic hierarchical connectivity. The first contribution is a precise definition based on a finite base set \(V_0\), an \(n\)-level supervertex family \(V\subseteq \mathcal{P}^n(V_0)\), a superedge family \(E\subseteq \mathcal{P}^{\ast}(V)\), a time domain \(T\), and an activity map \(\Lambda:E\to 2^T\). The second contribution is a hierarchy result showing that static \(n\)-Super-HyperGraphs, temporal hypergraphs, and temporal graphs are recovered by forgetting time or by imposing natural restrictions on \(n\) and edge cardinality. The third contribution is a collection of structural results proving that snapshots, time restrictions, temporal unions, temporal intersections, activity complements, and time shifts preserve the defining conditions of the model. The paper also presents a construction algorithm from static snapshots, proves its correctness, analyzes its complexity, and illustrates the interpretation of the model through project-management, logistics, and smart-building examples. These results give a rigorous mathematical basis for studying dynamic higher-order systems in which both temporal activation and hierarchical identity are essential.
