Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (non-quantum) areas. In this paper we survey these results and the quantum toolbox they use.

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Polynomial identity testing (PIT) is the problem of checking whether a given
arithmetic circuit is the zero circuit. PIT ranks as one of the most important
open problems in the intersection of algebra and computational complexity. In the last
few years, there has been an impressive progress on this ... more >>>

In the context of proving lower bounds on proof space in $k$-DNF
resolution, [Ben-Sasson and Nordström 2009] introduced the concept of
minimally unsatisfiable sets of $k$-DNF formulas and proved that a
minimally unsatisfiable $k$-DNF set with $m$ formulas can have at most
$O((mk)^{k+1})$ variables. They also gave an example of ... more >>>