  This paper develops a quantitative version of de Jong's central limit theorem
for homogeneous sums in a high-dimensional setting. More precisely, under
appropriate moment assumptions, we establish an upper bound for the Kolmogorov
distance between a multi-dimensional vector of homogeneous sums and a Gaussian
vector so that the bound depends polynomially on the logarithm of the dimension
and is governed by the fourth cumulants and the maximal influences of the
components. As a corollary, we obtain high-dimensional versions of fourth
moment theorems, universality results and Peccati-Tudor type theorems for
homogeneous sums. We also sharpen some existing (quantitative) central limit
theorems by applications of our result.
