  The lifetime of quasars is fundamental for understanding the growth of
supermassive black holes, and is an important ingredient in models of the
reionization of the intergalactic medium. However, despite various attempts to
determine quasar lifetimes, current estimates from a variety of methods are
uncertain by orders of magnitude. This work combines cosmological
hydrodynamical simulations and 1D radiative transfer to investigate the
structure and evolution of the He II Ly$\alpha$ proximity zones around quasars
at $z \simeq 3-4$. We show that the time evolution in the proximity zone can be
described by a simple analytical model for the approach of the He II fraction
$x_{\rm HeII}\left( t \right)$ to ionization equilibrium, and use this picture
to illustrate how the transmission profile depends on the quasar lifetime,
quasar UV luminosity, and the ionization state of helium in the ambient IGM
(i.e. the average He II fraction, or equivalently the metagalactic He II
ionizing background). A significant degeneracy exists between the lifetime and
the average He II fraction, however the latter can be determined from
measurements of the He II Ly$\alpha$ optical depth far from quasars, allowing
the lifetime to be measured. We advocate stacking existing He II quasar spectra
at $z\sim 3$, and show that the shape of this average proximity zone profile is
sensitive to lifetimes as long as $\sim 30$ Myr. At higher redshift $z\sim 4$
where the He II fraction is poorly constrained, degeneracies will make it
challenging to determine these parameters independently. Our analytical model
for He II proximity zones should also provide a useful description of the
properties of H I proximity zones around quasars at $z \simeq 6-7$.
