Beyond Talagrand functions: new lower bounds for testing monotonicity and unateness

We prove a lower bound of Ω(n1/3) for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f:{0,1}n→ {0,1} is monotone versus far from monotone. This improves the recent lower bound of Ω(n1/4) for the same problem by Belovs and Blais (STOC'16). Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity we prove a lower bound of Ω(√n) for two-sided, adaptive algorithms and a lower bound of Ω(n) for one-sided, non-adaptive algorithms for testing unateness, a natural generalization of monotonicity. The latter matches the linear upper bounds by Khot and Shinkar (RANDOM'16) and by Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova, and Seshadhri (2017).