  Understanding the growth of the supermassive black holes powering luminous
quasars, their co-evolution with host galaxies, and impact on the surrounding
intergalactic medium depends sensitively on the duration of quasar accretion
episodes. Unfortunately, this time-scale, known as the quasar lifetime, $t_{\rm
Q}$, is still uncertain by orders of magnitude ($t_{\rm Q}\simeq 0.01~{\rm
Myr}-1~{\rm Gyr}$). However, the extent of the He II Ly$\alpha$ proximity zones
in the absorption spectra of $z_{\rm qso}\sim3-4$ quasars constitutes a unique
probe, providing sensitivity to lifetimes up to $\sim 30$ Myr. Our recent
analysis of $22$ archival He II proximity zone spectra reveals a surprisingly
broad range of emission timescales, indicating that some quasars turned on
$\lesssim 1$ Myr ago, whereas others have been shining for $\gtrsim 30$ Myr.
Determining the underlying quasar lifetime distribution (QLD) from proximity
zone measurements is a challenging task owing to: 1) the limited sensitivity of
individual measurements; 2) random sampling of the quasar light curves; 3)
density fluctuations in the quasar environment; and 4) the inhomogeneous
ionization state of He II in a reionizing IGM. We combine a semi-numerical He
II reionization model, hydrodynamical simulations post-processed with ionizing
radiative transfer, and a novel statistical framework to infer the QLD from an
ensemble of proximity zone measurements. Assuming a log-normal QLD, we infer a
mean $\langle {\rm log}_{10}\left(t_{\rm Q}/{\rm
Myr}\right)\rangle=0.22^{+0.22}_{-0.25}$ and standard deviation $\sigma_{{\rm
log}_{10}t_{\rm Q}}=0.80^{+0.37}_{-0.27}$. Our results allow us to estimate the
probability of detecting young quasars with $t_{\rm Q}\leq0.1$ Myr from their
proximity zone sizes yielding $p\left(\leq 0.1~{\rm
Myr}\right)=0.19^{+0.11}_{-0.09}$, which is broadly consistent with recent
determination at $z\sim 6$.
