Which results make quantum space interesting?
I think Amnon Ta-Shma's new result is a good answer to my own question.


Amnon Ta-Shma: Inverting well conditioned matrices in quantum logspace. STOC 2013: 881-890. (copy from author's website)



  ...  We show that the inverse of a well conditioned matrix can be approximated in quantum logspace with intermediate measurements. This should be compared with the best known classical algorithm for the problem that requires $\Omega(\log^2 n)$ space. We also show how to approximate the spectrum of a normal matrix, or the singular values of an arbitrary matrix, with $\epsilon$ additive accuracy, and how to approximate the singular value decomposition (SVD) of a matrix whose singular values are well separated. ...