We prove an Ω(dlg n/(lg lg n) ²) lower bound on the dynamic cell-probe complexity of statistically oblivious approximate-near-neighbor search (ANN) over the ddimensional Hamming cube. For the natural setting of d = Θ(lg n), our result implies an Ω(lg ² n) lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for ANN. This is the first super-logarithmic unconditional lower bound for ANN against general (non black-box) data structures. We also show that any oblivious static data structure for decomposable search problems (like ANN) can be obliviously dynamized with O(lg n) overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).