I know the circuit depth (i.e. amount of sequential operations on the longest path) for long multiplication can easily be made $O(\log(n))$ (possibly at the cost of increasing the computational complexity by a small constant factor), and I’m pretty sure the same applies to Karatsuba and Toom-Cook. However, I don’t understand enough about the asymptotically faster algorithms (e.g. Schönhage–Strassen or faster) to know their circuit depth. Are they also $O(\log(n))$ or is there something inherently sequential about them, effectively creating a complexity-parallelizability trade off.