This work explores the Tate profile and its applications. We show that, framed using Tate profiles, several well-known results on divisibility and basis sampling become natural. Then, having rewritten these results in this more natural setting, we generalize and optimize several use cases: First, we use Tate profiles to generalize entangled basis generation to any Weierstrass curve. Second, we optimize the computation of degree-2 Tate pairings on Kummer surfaces using cubical arithmetic, which allows us to compute the 2-Tate profile in only 6 additions, 10 multiplications, and 4 Legendre symbols. Third, we apply the optimized 2-Tate profile computation to perform subgroup membership testing on the Gaudry–Schost Kummer Surface, smoothly combining these ideas.