We report Atacama Large Millimeter/submillimeter Array (ALMA) observations of CO $(8-7)$, $(9-8)$, $rm H_{2}O (2_{0,2}-1_{1,1})$ and $rm OH^{+} (1_{1}-0_{1})$ and NOrthern Extended Millimeter Array (NOEMA) observations of CO $(5-4)$, $(6-5)$, $(12-11)$ and $(13-12)$ towards the $z = 6.003$ quasar SDSS J231038.88+185519.7, aiming to probe the physical conditions of the molecular gas content of this source. We present the best sampled CO spectral line energy distribution (SLED) at $z = 6.003$, and analyzed it with the radiative transfer code MOLPOP-CEP. Fitting the CO SLED to a one-component model indicates a kinetic temperature $T_{rm kin} = 228 rm K$, molecular gas density $log (n(rm H_{2})/rm cm^{-3}$ )=4.75, and CO column density $log(N(rm CO)/rm cm^{-2}) =17.5$, although a two-component model better fits the data. In either case, the CO SLED is dominated by a warm and dense component. Compared to samples of local (Ultra) Luminous Infrared Galaxies ((U)LIRGs), starburst galaxies and high redshift Submillimeter Galaxies (SMGs), J2310+1855 exhibits higher CO excitation at ($J geq 8$), like other high redshift quasars. The high CO excitation, together with the enhanced $L_{rm H_{2}O}/ L_{IR} $, $L_{rm H_{2}O}/ L_{CO} $ and $L_{OH^{+}}/L_{rm H_{2}O} $ ratios, suggests that besides the UV radiation from young massive stars, other mechanisms such as shocks, cosmic rays and X-rays might also be responsible for the heating and ionization of the molecular gas. In the nuclear region probed by the molecular emissions lines, any of these mechanisms might be present due to the powerful quasar and the starburst activity.
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