  Proximity zones of high-redshift quasars are unique probes of their central
supermassive black holes as well as the intergalactic medium in the last stages
of reionization. We present 22 new measurements of proximity zones of quasars
with redshifts between 5.8 and 6.6, using the enlarged XQR-30 sample of
high-resolution, high-SNR quasar spectra. The quasars in our sample have UV
magnitudes of $M_{1450}\sim -27$ and black hole masses of
$10^9$$\unicode{x2013}$$10^{10}$ M$_\odot$. Our inferred proximity zone sizes
are 2$\unicode{x2013}$7 physical Mpc, with a typical uncertainty of less than
0.5 physical Mpc, which, for the first time, also includes uncertainty in the
quasar continuum. We find that the correlation between proximity zone sizes and
the quasar redshift, luminosity, or black hole mass, indicates a large
diversity of quasar lifetimes. Two of our proximity zone sizes are
exceptionally small. The spectrum of one of these quasars, with $z=6.02$,
displays, unusually for this redshift, damping wing absorption without any
detectable metal lines, which could potentially originate from the IGM. The
other quasar has a high-ionization absorber $\sim$0.5 pMpc from the edge of the
proximity zone. This work increases the number of proximity zone measurements
available in the last stages of cosmic reionization to 87. This data will lead
to better constraints on quasar lifetimes and obscuration fractions at high
redshift, which in turn will help probe the seed mass and formation redshift of
supermassive black holes.
