Randomized versus quantum query complexity of total Boolean functions: is R(f) = O(Q(f)^3)?

Theorem 1.1: for each fixed $k\ge2$ there are total Boolean functions $F_{k,m}$ with $Q(F_{k,m})\le C_k\sqrt m(\log m)^{b_k}$ and $R(F_{k,m})\ge c_km^{2-1/k}/(\log m)^2$, a power $4-2/k$ separation up to logarithms, so no bound $R=O((1+Q)^\alpha)$ holds with $\alpha<4$. With the known $R=O(Q^4)$ the optimal exponent is exactly 4, and the cubic conjecture is false. The functions are cheat-sheet-style columns whose certificate cells are addressed by answers to partial-function instances. Not shown: a separation with exponent exactly 4 for a single family, or new results for partial functions.

Theorem 1.1: for each fixed $k\ge2$ there are total Boolean functions $F_{k,m}$ with $Q(F_{k,m})\le C_k\sqrt m(\log m)^{b_k}$ and $R(F_{k,m})\ge c_km^{2-1/k}/(\log m)^2$, a power $4-2/k$ separation up to logarithms, so no bound $R=O((1+Q)^\alpha)$ holds with $\alpha<4$. With the known $R=O(Q^4)$ the optimal exponent is exactly 4, and the cubic conjecture is false. The functions are cheat-sheet-style columns whose certificate cells are addressed by answers to partial-function instances. Not shown: a separation with exponent exactly 4 for a single family, or new results for partial functions.