  Recently, two classes of quasar samples were identified, which are promising
as new cosmological probes extending to higher redshifts. The first sample uses
the nonlinear relation between the ultraviolet and X-ray luminosities of
quasars to derive luminosity distances, whereas the linear sizes of compact
radio quasars in the second sample can serve as standardized rulers, providing
angular-diameter distances. In this study, under the assumption of a flat
universe, we refreshed the calibration of multiple measurements of
high-redshift quasars (in the framework of a cosmological-model-independent
method with the newest Hubble parameters data). Furthermore, we placed
constraints on four models that characterize the cosmic equation of state
($w$). The obtained results show that: 1) the two quasar samples could provide
promising complementary probes at much higher redshifts, whereas compact radio
quasars perform better than ultraviolet and X-ray quasars at the current
observational level; 2) strong degeneracy between the cosmic equation of state
($w$) and Hubble constant ($H_0$) is revealed, which highlights the importance
of independent determination of $H_0$ from time-delay measurements of strongly
lensed Quasars; 3)together with other standard ruler probes, such as baryon
acoustic oscillation distance measurements, the combined QSO+BAO measurements
are consistent with the standard $\Lambda$CDM model at a constant equation of
state $w=-1$; 4) ranking the cosmological models, the polynomial
parametrization gives a rather good fit among the four cosmic-equation-of-state
models, whereas the Jassal-Bagla-Padmanabhan (JBP) parametrization is
substantially penalized by the Akaike Information Criterion and Bayesian
Information Criterion criterion.
