In this talk, I will present a new method that reduces understanding an appropriate notion of girth in hypergraphs to unraveling the spectrum of a "Kikuchi" matrix associated with the hypergraph. I will discuss three applications of this technique: 1. Finding a refutation algorithm for smoothed instances of constraint satisfaction problems (obtained by randomly perturbing the literal patterns in a worst-case instance with a small probability) that matches the best running-time vs constraint-density trade-offs for the significantly special and easier case of random CSPs, 2. Confirming Feige's 2008 Conjecture that postulated an extremal girth vs density trade-off (a.k.a. Moore bounds) for k-uniform hypergraphs that generalizes the Alon-Hoory-Linial Moore bound for graphs, 3. Proving a cubic lower bound on the block length of 3 query locally decodable codes improving on the prior best quadratic lower bound from the early 2000s. Based on joint works with Omar Alrabiyah (Berkeley), Tim Hsieh (CMU), Peter Manohar (CMU), Sidhanth Mohanty (Berkeley), and Venkat Guruswami (Berkeley).