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Cool gas (T$\sim$10$^{4}$~K) traced by hydrogen Ly$\alpha$ emission is now routinely detected around $z\sim3$ quasars, but little is known about their molecular gas reservoirs. Here, we present an APEX spectroscopic survey of the CO(6-5), CO(7-6) and [CI](2-1) emission lines for 9 quasars from the QSO MUSEUM survey 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, we constrain the molecular gas reservoirs to be $\rm M_{H_{2}} = (0.4-6.9) \times 10^{11} M_{\odot}$, while the non-detections imply $\rm M_{H_{2}} < 1.1\times 10^{11} M_{\odot}$. We compare our observations with the extended Ly$\alpha$ properties to understand the link between the cool and the molecular gas phases. We find large velocity shifts between the bulk of Ly$\alpha$ and the molecular gas systemic redshift in five sources (from $\sim$-400 to $\sim+$1200~km~s$^{-1}$). The sources with the largest shifts have the largest Ly$\alpha$ line widths in the sample, suggesting more turbulent gas conditions and/or large-scale inflows/outflows around these quasars. We also find that the brightest ($I_{\rm[CI](2-1)}=15.7\pm3.7~\rm Jy~km~s^{-1}$) and the widest (FWHM$\sim$900~km~s$^{-1}$) lines are detected for the smallest and dimmest Ly$\alpha$ nebulae. From this, we speculate that host galaxy obscuration can play an important role in reducing the ionizing and Ly$\alpha$ photons able to escape to halo scales, and/or that these systems are hosted by more massive halos., Comment: Accepted on January 3, MNRAS. 25 pages, 17 figures

Context. The formation and evolution of galaxies appear linked to the growth of supermassive black holes, as evidenced by empirical scaling relations in nearby galaxies. Understanding this co-evolution over cosmic time requires the revelation of the dynamical state of galaxies and the measurement of the mass of their central black holes (MBH) at a range of cosmic distances. Bright active galactic nuclei (AGNs) are ideal for this purpose. Aims. The High Angular Resolution Monolithic Optical and Near-infrared Integral field spectrograph (HARMONI), the first light integral-field spectrograph for the Extremely Large Telescope (ELT), will transform visible and near-infrared ground-based astrophysics thanks to its advances in sensitivity and angular resolution. We aim to analyse the capabilities of HARMONI to reveal the stellar morpho-kinematic properties of the host galaxies of AGNs at about cosmic noon. Methods. We made use of the simulation pipeline for HARMONI (HSIM) to create mock observations of representative AGN host galaxies at redshifts around cosmic noon. We used observations taken with the Multi-Unit Spectroscopic Explorer (MUSE) of nearby galaxies showing different morphologies and dynamical stages combined with theoretical AGN spectra to create the target inputs for HSIM. Results. According to our simulations, an on-source integration time of three hours should be enough to measure the MBH and to trace the morphology and stellar kinematics of the brightest host galaxies of AGNs beyond cosmic noon. For host galaxies with stellar masses $, Comment: 30 pages, 28 figures, accepted for publication in A&A