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We show nearly quadratic separations between two pairs of complexitymeasures:  1. We show that there is a Boolean function $f$ with$D(f)=\Omega((D^{sc}(f))^{2-o(1)})$ where $D(f)$ is the deterministic querycomplexity of $f$ and $D^{sc}$ is the subcube partition complexity of $f$;  2. As a consequence, we obtain that there is a communication task $f(x, y)$such that $D^{cc}(f)=\Omega(\log^{2-o(1)}\chi(f))$ where $D^{cc}(f)$ is thedeterministic 2-party communication complexity of $f$ (in the standard 2-partymodel of communication) and $\chi(f)$ is the partition number of $f$.  Both of those separations are nearly optimal: it is well known that$D(f)=O((D^{sc}(f))^{2})$ and $D^{cc}(f)=O(\log^2\chi(f))$.

Almost quadratic gap between partition complexity and query/communication complexity