  Gauge boson self-couplings are completely defined by the non-Abelian gauge
symmetry of the Standard Model (SM). For this reason, a direct search for these
couplings is extremely significant in understanding the gauge structure of the
SM. The possible deviation from the SM predictions of gauge boson
self-couplings would be a sign of the presence of new physics beyond the SM. In
this work, we study the sensitivities on the anomalous couplings defined by
dimension-8 operators related to the $ZZ\gamma\gamma$ and $Z\gamma\gamma\gamma$
quartic vertices through the process $e^+e^- \to Z\gamma\gamma$ with Z-boson
decaying to charged leptons at the Compact Linear Collider (CLIC). We analyze
the center-of-mass energy of 3 TeV, integrated luminosities of ${\cal L}=1, 4,
5$ $\rm ab^{-1}$, systematic uncertainties of $\delta_{sys}=0\%, 3\%, 5\%$,
unpolarized and polarized electron beams and considering the Initial State
Radiation and Beamstrahlung effects for extraction of expected sensitivity on
the anomalous $f_ {T,j}/\Lambda^4$ couplings at $95\%$ confidence level, which
are especially sensitive to the $Z\gamma\gamma$ channel. It is clear from the
results that we expect better limits on the couplings if the systematic error
is improved. The best limits obtained on the anomalous quartic couplings for
the process $e^+e^- \to Z\gamma\gamma$ with $\sqrt{s}= 3$ TeV, ${\cal L}=5$
$\rm ab^{-1}$, and $\delta_{sys}=0\%$ can be approximately improved up to about
1.1 times better than the limits obtained with $\delta_{sys}=5\%$. Our
sensitivities on the anomalous quartic couplings can set more stringent
sensitivity by two orders of magnitude concerning the best sensitivity derived
from the current experimental limits. Finally, with initial electron beam
polarization, the sensitivity of the anomalous quartic couplings improves by
almost a factor of 1.2.
