  Constraints on the time-scales of quasar activity are key to understanding
the formation and growth of supermassive black holes (SMBHs), quasar triggering
mechanisms, and possible feedback effects on their host galaxies. However,
observational estimates of this so-called quasar lifetime are highly uncertain
(t_Q~10^4-10^9 yr), because most methods are indirect and involve many
model-dependent assumptions. Direct evidence of earlier activity is gained from
the higher ionization state of the intergalactic medium (IGM) in the quasar
environs, observable as enhanced Ly$\alpha$ transmission in the so-called
proximity zone. Due to the ~30 Myr equilibration time-scale of HeII in the z~3
IGM, the size of the HeII proximity zone depends on the time the quasar had
been active before our observation t_on<t_Q, enabling up to $\pm$0.2 dex
precise measurements of individual quasar on-times that are comparable to the
e-folding time-scale t_S~44 Myr of SMBH growth. Here we present the first
statistical sample of 13 accurate and precise measurements of HeII quasar
proximity zone sizes from science-grade (signal-to-noise ratio ~3) Hubble Space
Telescope (HST) spectra, spanning ~2 to ~15 proper Mpc. Comparing these sizes
to predictions from cosmological hydrodynamical simulations post-processed with
one-dimensional radiative transfer, we infer a broad range of quasar on-times
from t_on<1 Myr to t_on>30 Myr that does not depend on quasar luminosity, black
hole mass, or Eddington ratio. These results point to episodic quasar activity
over a long duty cycle, but do not rule out substantial SMBH growth during
phases of radiative inefficiency or obscuration.
