If $X,Y$ are two independent random variables with CDFs $F_X,F_Y$, their sum has CDF $F_X \star F_Y$ ($\star$ is the convolution product).
What can be said about the quantile function of $X+Y$ ? The quantile function should be the inverse of $F_X \star F_Y$. Can we express that using the CDFs, quantiles (or even densities) of $X$ and $Y$ ?