  This work studies the estimation of many statistical quantiles under
differential privacy. More precisely, given a distribution and access to i.i.d.
samples from it, we study the estimation of the inverse of its cumulative
distribution function (the quantile function) at specific points. For instance,
this task is of key importance in private data generation. We present two
different approaches. The first one consists in privately estimating the
empirical quantiles of the samples and using this result as an estimator of the
quantiles of the distribution. In particular, we study the statistical
properties of the recently published algorithm introduced by Kaplan et al. 2022
that privately estimates the quantiles recursively. The second approach is to
use techniques of density estimation in order to uniformly estimate the
quantile function on an interval. In particular, we show that there is a
tradeoff between the two methods. When we want to estimate many quantiles, it
is better to estimate the density rather than estimating the quantile function
at specific points.
