Adaptivity vs Postselection

We study the following problem: with the power of postselection (classically or quantumly), what is your ability to answer adaptive queries to certain languages? More specifically, for what kind of computational classes $\mathcal{C}$, we have $\mathsf{P}^{\mathcal{C}}$ belongs to $\mathsf{PostBPP}$ and $\mathsf{PostBQP}$? While a complete answer to the above question seems impossible given the development of present computational complexity theory. We study the analogous question in query complexity, which sheds light on the limitation of {\em relativized} methods (the relativization barrier) to the above question. Informally, we show that, for a partial function $f$, if there is no efficient (In the world of query complexity, being efficient means using $O(\operatorname*{polylog}(n))$ time.) {\em small bounded-error} algorithm for $f$ classically or quantumly, then there is no efficient postselection bounded-error algorithm to answer adaptive queries to $f$ classically or quantumly. Our results imply a new proof for the classical oracle separation $\mathsf{P}^{\mathsf{NP}^{\mathcal{O}}} \not\subset \mathsf{PP}^{\mathcal{O}}$. They also lead to a new oracle separation $\mathsf{P}^{\mathsf{SZK}^{\mathcal{O}}} \not\subset \mathsf{PP}^{\mathcal{O}}$. Our result also implies a hardness amplification construction for polynomial approximation: given a function $f$ on $n$ bits, we construct an adaptive-version of $f$, denoted by $F$, on $O(m \cdot n)$ bits, such that if $f$ requires large degree to approximate to error $2/3$ in a certain one-sided sense, then $F$ requires large degree to approximate even to error $1/2 - 2^{-m}$. Our construction achieves the same amplification in the work of Thaler (ICALP, 2016), by composing a function with $O(\log n)$ {\em deterministic query complexity}.