Quantum proofs of classical theorems I'm interested in examples of problems where a theorem which seemingly has nothing to do with quantum mechanics/information (e.g. states something about purely classical objects) can nevertheless be proved using quantum tools. A survey Quantum Proofs for Classical Theorems (A. Drucker, R. Wolf) gives a nice list of such problems, but surely there are many more. 

Particularly interesting would be examples where a quantum proof is not only possible, but also "more illuminating", in analogy with real and complex analysis, where putting a real problem in the complex setting often makes it more natural (e.g. geometry is simpler since $\mathbb{C}$ is algebraically closed etc.); in other words, classical problems for which quantum world is their "natural habitat".

(I'm not defining "quantumness" here in any precise sense and one could argue that all such arguments eventually boil down to linear algebra; well, one can also translate any argument using complex numbers to use only pairs of reals - but so what?)
There is a recent paper from  Scott Aaronson which provides a new proof that the permanent is #P-hard. This proof is based on the model of linear-optical quantum computing and is more intuitive than that of Leslie Valiant.