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Cool gas (T$\sim$10$^{4}$~K) traced by hydrogen Ly$\alpha$ emission is nowroutinely detected around $z\sim3$ quasars, but little is known about theirmolecular gas reservoirs. Here, we present an APEX spectroscopic survey of theCO(6-5), CO(7-6) and [CI](2-1) emission lines for 9 quasars from the QSO MUSEUMsurvey which have similar UV luminosities, but very diverse Ly$\alpha$ nebulae.These observations ($\langle~\rm rms~\rangle=2.6$~mJy in 300~km~s$^{-1}$)detected three CO(6-5) lines with 3.4$\leq I_{\rm CO(6-5)}\leq$5.1~Jy~km~s$^{-1}$, 620$\leq$FWHM$\leq$707~km~s$^{-1}$, and three[CI](2-1) lines with 2.3$\leq I_{\rm [CI](2-1)} \leq$15.7~Jy~km~s$^{-1}$,329$\leq$FWHM$\leq$943~km~s$^{-1}$. For the CO and [CI] detected sources, weconstrain the molecular gas reservoirs to be $\rm M_{H_{2}} = (0.4-6.9) \times10^{11} M_{\odot}$, while the non-detections imply $\rm M_{H_{2}} < 1.1\times10^{11} M_{\odot}$. We compare our observations with the extended Ly$\alpha$properties to understand the link between the cool and the molecular gasphases. We find large velocity shifts between the bulk of Ly$\alpha$ and themolecular gas systemic redshift in five sources (from $\sim$-400 to$\sim+$1200~km~s$^{-1}$). The sources with the largest shifts have the largestLy$\alpha$ line widths in the sample, suggesting more turbulent gas conditionsand/or large-scale inflows/outflows around these quasars. We also find that thebrightest ($I_{\rm[CI](2-1)}=15.7\pm3.7~\rm Jy~km~s^{-1}$) and the widest(FWHM$\sim$900~km~s$^{-1}$) lines are detected for the smallest and dimmestLy$\alpha$ nebulae. From this, we speculate that host galaxy obscuration canplay an important role in reducing the ionizing and Ly$\alpha$ photons able toescape to halo scales, and/or that these systems are hosted by more massivehalos.

APEX at the QSO MUSEUM: molecular gas reservoirs associated with $z\sim$3 quasars and their link to the extended Ly$\alpha$ emission