In principle, the curse of dimensionality kicks in after a dozen or so dimensions, as convergence is O( log(N)^D / N). But in practice I have found that in a surprising number of applications and circumstances, quasirandom sequences can have offer a substantial improvement even when the number of dimensions are in the hundreds or even thousands. (Some authors have suggested this works because the solution space is of low dimensions but embedded in a much higher dimensional space...)
However, to get the full benefit I needed to find a low discrepancy quasirandom sequence that did not suffer the many of the parameter degeneracy problems that many of the conventional ones exhibit for very large D. For me, the new R-sequence nicely solves this parameter selection problem by not having any parameters to optimize!