Modeling complex, multi-way interactions with hypergraphs presents a fundamental tradeoff between mathematical expressivity and computational scalability. In this talk, we demonstrate how implicit approaches and randomized algorithms serve as critical tools for mitigating this bottleneck. Specifically, we highlight recurring challenges in scaling hypergraph matrix and tensor methods for large datasets. For hypergraph matrices and Laplacians, we identify prohibitive density as a primary barrier to downstream analysis. We show how random sampling accurately predicts this density, and how subsequent implicit approaches enable efficient hypergraph matrix sparsification, construction, and spectral analysis. Moving to hypergraph tensors, we address the dual challenges of hyperedge nonuniformity and high-dimensionality. We introduce an implicit algorithm for executing a core primitive — tensor-times-same-vector (TTSV) — on a nonuniform hypergraph adjacency tensor without ever requiring its explicit formation. Through concrete examples, we illustrate how these scalable tensor methods capture structural nuances that remain invisible to matrix-based approaches. We will conclude with a discussion of open problems and future directions.