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We investigate whether the hard X-ray photon index ($Γ$) versus accretion rate correlation for super-Eddington accreting quasars is different from that for sub-Eddington accreting quasars. We construct a sample of 113 bright quasars from the Sloan Digital Sky Survey Data Release 14 quasar catalog, including 38 quasars as the super-Eddington subsample and 75 quasars as the sub-Eddington subsample. We derive black-hole masses using a simple-epoch virial mass formula based on the ${\rm Hβ}$ lines, and we use the standard thin disk model to derive the dimensionless accretion rates ($\dot{\mathscr{M}}$) for our sample. The X-ray data for these quasars are collected from the Chandra and XMM-Newton archives. We fit the hard X-ray spectra using a single power-law model to obtain $Γ$ values. We find a statistically significant ($R_{\rm S}=0.43$, $p=7.75\times{10}^{-3}$) correlation between $Γ$ and $\dot{\mathscr{M}}$ for the super-Eddington subsample. The $Γ$-$\dot{\mathscr{M}}$ correlation for the sub-Eddington subsample is also significant, but weaker ($R_{\rm S}=0.30$, $p=9.98\times{10}^{-3}$). Linear regression analysis shows that ${\rm Γ}=(0.34\pm0.11){\rm log}{\dot{\mathscr{M}}}+(1.71\pm0.17)$ and ${\rm Γ}=(0.09\pm0.04){\rm log}{\dot{\mathscr{M}}}+(1.93\pm0.04)$ for the super- and sub-Eddington subsamples, respectively. The $Γ$-$\dot{\mathscr{M}}$ correlations of the two subsamples are different, suggesting different disk-corona connections in these two types of systems. We propose one qualitative explanation of the steeper $Γ$-$\dot{\mathscr{M}}$ correlation in the super-Eddington regime that involves larger seed photon fluxes received by the compact coronae from the thick disks in super-Eddington accreting quasars.