On polynomially many queries to NP or QMA oracles

Abstract: We study the complexity of problems solvable in deterministic polynomial time with access to an NP or Quantum Merlin-Arthur (QMA)-oracle, such as $P<sup>{NP}$ and $P<sup>{QMA}$, respectively. The former allows one to classify problems more finely than the Polynomial-Time Hierarchy (PH), whereas the latter characterizes physically motivated problems such as Approximate Simulation (APX-SIM) [Ambainis, CCC 2014]. In this area, a central role has been played by the classes $P<sup>{NP[\log]}$ and $P<sup>{QMA[\log]}$, defined identically to $P<sup>{NP}$ and $P<sup>{QMA}$, except that only logarithmically many oracle queries are allowed. Here, [Gottlob, FOCS 1993] showed that if the adaptive queries made by a $P<sup>{NP}$ machine have a "query graph" which is a tree, then this computation can be simulated in $P<sup>{NP[\log]}$. In this work, we first show that for any verification class $C\in{NP,MA,QCMA,QMA,QMA(2),NEXP,QMA_{\exp}}$, any $P<sup>C$ machine with a query graph of "separator number" $s$ can be simulated using deterministic time $\exp(s\log n)$ and $s\log n$ queries to a $C$-oracle. When $s\in O(1)$ (which includes the case of $O(1)$-treewidth, and thus also of trees), this gives an upper bound of $P<sup>{C[\log]}$, and when $s\in O(\log<sup>k(n))$, this yields bound $QP<sup>{C[\log<sup>{k+1}]}$ (QP meaning quasi-polynomial time). We next show how to combine Gottlob's "admissible-weighting function" framework with the "flag-qubit" framework of [Watson, Bausch, Gharibian, 2020], obtaining a unified approach for embedding $P<sup>C$ computations directly into APX-SIM instances in a black-box fashion. Finally, we formalize a simple no-go statement about polynomials (c.f. [Krentel, STOC 1986]): Given a multi-linear polynomial $p$ specified via an arithmetic circuit, if one can "weakly compress" $p$ so that its optimal value requires $m$ bits to represent, then $P<sup>{NP}$ can be decided with only $m$ queries to an NP-oracle.