Summary: ``Let $(X,Y)$ be a bivariate random vector and $F(x)$ the marginal distribution function of $X$. The quantile regression (QR) function of $Y$ on $X$ is defined as $r(u)={\bf E}[Y|F(X)=u]$ and the cumulative QR function (CQR) $M(u)$ as its integral over $[0,u]$. The empirical counterpart based on a sample of size $n$ is $M_n(u)$. In this paper, we construct strong Gaussian approximations of the associated CQR process under appropriate assumptions. The construction provides a firm basis for the study of functional statistics based on $M_n(u)$. A law of the iterated logarithm for the CQR process follows from our result.''