Is there a quantum NC algorithm for computing GCD? From the comments on one of my questions on MathOverflow 
I get the feeling that the question regarding GCD being in $\mathsf{NC}$ vs. $\mathsf{P}$ is akin to the question regarding Integer Factorization being in $\mathsf{P}$ vs. $\mathsf{NP}$. 

Is there something like a "quantum $\mathsf{NC}$" algorithm for GCD as there is a quantum polynomial time ($\mathsf{BQP}$) algorithm for Integer Factorization?

Related question: complexity of greatest common divisor (gcd)
First of all, there is a formal definition of "quantum-NC", see QNC on the zoo.

GCD is indeed a good candidate for a problem that could be shown to be in QNC, but it's not known to be in NC. However, finding a QNC algorithm for GCD is still an open problem. 

The feeling for which this is believed to be true comes from the fact that the Quantum Fourier Transform can be done in QNC.

Reference: Conclusion section of "R. Cleve and J. Watrous, Fast parallel circuits for the quantum Fourier transform", arXiv:quant-ph/0006004