TITLE: God's number for the $n \times n \times n$-cube
QUESTION [13 upvotes]: This is a question about Rubik's Cube and generalizations of this puzzle, such as Rubik's Revenge, Professor's cube or in general the $n \times n \times n$ cube.
Let $g(n)$ be the smallest number $m$, such that every realizable arrangement of the $n \times n \times n$ cube can be solved with $m$ moves. In other words, this is the "radius" of the Cayley graph of the $n \times n \times n$ cube group with respect to the canonical generating system.
We have $g(1)=0$, $g(2)=11$ and - quite recently - in 2010 it was proven that $g(3)=20$: God's number is 20.
Question. Is anything known about $g(4)$ or $g(5)$?
I expect that the precise number is unkwown since the calculation for Rubik's cube already took three decades. Nevertheless, is there any work in progress? Are any lower or upper bounds known?
I would like to ask the same question about $g(n)$ for $n>5$, or rather:
Question. Is anything known about the asymptotic value of $g(n)$?

REPLY [5 votes]: Based on this discussion from 2015, God's number for the 4x4x4 cube lies beween 35 and 55 (inclusive) for the outer block turn metric, with the corresponding bounds being 32 and 53 for the single slice turn metric, and 29 and 53 for the block turn metric. Scroll down to the comments for the optimal bounds.
The choice of metric is irrelevant for the asymptotic results of Demaine, Demaine, Eisenstat, Lubiw and Winslow which gives $\Theta(n^2/\log(n))$ as the answer to question two.
Since this question was first asked, cube20.org claims to have also shown that God's number for the quarter turn metric is 26.
As for the 5x5x5, one can find this post claiming an upper bound of 130 for the outer block turning metric.