  We present a new model-independent method to determine the spatial curvature
and to mitigate the circularity problem affecting the use of quasars as
distance indicators. The cosmic-chronometer measurements are used to construct
the curvature-dependent luminosity distance $D^{\rm cal}_{L}(\Omega_{K},z)$
using a polynomial fit. Based on the reconstructed $D^{\rm
cal}_{L}(\Omega_{K},z)$ and the known ultraviolet versus X-ray luminosity
correlation of quasars, we simultaneously place limits on the curvature
parameter $\Omega_{K}$ and the parameters characterizing the luminosity
correlation function. This model-independent analysis suggests that a mildly
closed Universe ($\Omega_{K}=-0.918\pm0.429$) is preferred at the $2.1\sigma$
level. With the calibrated luminosity correlation, we build a new data set
consisting of 1598 quasar distance moduli, and use these calibrated
measurements to test and compare the standard $\Lambda$CDM model and the
$R_{\rm h}=ct$ universe. Both models account for the data very well, though the
optimized flat $\Lambda$CDM model has one more free parameter than $R_{\rm
h}=ct$, and is penalized more heavily by the Bayes Information Criterion. We
find that $R_{\rm h}=ct$ is slightly favoured over $\Lambda$CDM with a
likelihood of $\sim57.7\%$ versus 42.3\%.
