  Quasi-Monte Carlo (QMC) methods for high dimensional integrals over unit
cubes and products of spheres are well-studied in literature. We study QMC
tractability of integrals of functions defined over the product of $m$ copies
of the simplex $T^d \subset \mathbb{R}^{d}$. The domain is a tensor product of
$m$ reproducing kernel Hilbert spaces defined by `weights' $\gamma_{m,j}$, for
$j = 1,2, \ldots, m$. Similar to the results on the unit cube in $m$
dimensions, and the product of $m$ copies of the $d$-dimensional sphere, we
prove that strong polynomial tractability holds iff $\limsup_{m \rightarrow
\infty} \sum_{j=1}^m \gamma_{m,j} < \infty$ and polynomial tractability holds
iff $\limsup_{m \rightarrow \infty} \frac{\sum_{j=1}^m \gamma_{m,j}}{\log(m + 1
)} < \infty$. We also show that weak tractability holds iff $\lim_{m
\rightarrow \infty} \frac{\sum_{j=1}^m \gamma_{m,j}}{m} = 0$. The proofs employ
Sobolev space techniques and weighted reproducing kernel Hilbert space
techniques for the simplex and products of simplices as domain. Properties of
orthogonal polynomials on a simplex are also used extensively.
