What is Low-discrepancy sequence, Sobol sequence, quasi-Monte Carlo?

The star discrepancy $D_n^*$ of points $x_1, \dots, x_n$ in $[0, 1)^d$ is the largest difference, over boxes $[0, a)$, between the fraction of points in the box and its volume. A low-discrepancy sequence has $D_n^* = O((\log n)^d/n)$. The Sobol sequence (Sobol, 1967) is a base-2 low-discrepancy sequence built coordinate by coordinate from primitive polynomials and direction numbers, by exclusive-or of binary digits. Quasi-Monte Carlo replaces the random draws of Monte Carlo by such points.