Abstract: Persistent homology provides a robust way to extract topological signatures from data, but its direct computation can become expensive for large complexes and filtrations. In this talk, I will discuss two complementary directions: first, how edge collapse can substantially simplify flag filtrations while preserving persistent homology, through the swap, shift, and trim operations; and second, how persistent homology can be used to classify temporal graphs by converting temporal motifs into filtrations. The talk will highlight both the algorithmic gains from collapse-based preprocessing and the effectiveness of topological signatures in temporal graph classification, with comparisons against graph-filtration kernels baselines.