Sources at the brightest end of QSO luminosity function during the peak epochof star formation and black hole accretion (z~2-4, i.e. Cosmic noon) areprivileged sites to study the feeding & feedback cycle of massive galaxies. Weperform the first systematic study of cold gas properties in the most luminousQSOs, by characterising their host-galaxies and environment. We analyse ALMA,NOEMA and JVLA observations of FIR continuum, CO and [CII] emission lines ineight QSOs ($L_{\rm Bol}>3\times10^{47}$ erg/s) from the WISSH sample atz~2.4-4.7. We report a 100% emission line detection rate and a 80% detectionrate in continuum emission, and we find CO emission to be consistent with thesteepest CO ladders observed so far. Sub-mm data reveal presence of (one ormore) bright companion galaxies around 80% of WISSH QSOs, at projecteddistances of 6-130 kpc. We observe a variety of sizes for the molecular gasreservoirs (1.7-10 kpc), associated with rotating disks with disturbedkinematics. WISSH QSOs typically show lower CO luminosity and higher starformation efficiency than FIR matched, z~0-3 main-sequence galaxies, implyingthat, given the observed SFR ~170-1100 $M_\odot$/yr, molecular gas is convertedinto stars on <50 Myr. Most targets show extreme dynamical to black-hole massratios $M_{\rm dyn}/M_{\rm BH}\sim3-10$, two orders of magnitude smaller thanlocal relations. The molecular gas fraction in WISSH hosts is lower by a factorof ~10-100 than in star forming galaxies with similar $M_*$. WISSH QSOs undergoan intense growth phase of both the central SMBH and host-galaxy. They pinpointhigh-density sites where giant galaxies assemble and mergers play a major rolein the build-up of the final host-galaxy mass. The observed low molecular gasfraction and short depletion timescale are likely due to AGN feedback, astraced by fast AGN-driven ionised outflows in all our targets.

We resolve the host galaxies of seven gravitationally lensed quasars atredshift 1.5 to 2.8 using observations with the Atacama Large (sub-)MillimetreArray. Using a visibility-plane lens modelling technique, we create pixellatedreconstructions of the dust morphology, and CO line morphology and kinematics.We find that the quasar hosts in our sample can be distinguished into twotypes: 1) galaxies characterised by clumpy, extended dust distributions($R_{\rm eff}\sim2$ kpc) and mean star formation rate surface densitiescomparable to sub-mm-selected dusty star-forming galaxies ($\Sigma_{\rmSFR}\sim3$ M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$); 2) galaxies that have sizes indust emission similar to coeval passive galaxies and compact starbursts($R_{\rm eff}\sim0.5$ kpc), with high mean star formation rate surfacedensities ($\Sigma_{\rm SFR}=$ 400$-$4500 M$_{\odot}$ yr$^{-1}$ kpc$^{-2}$)that may be Eddington-limited or super-Eddington. The small size of some quasarhosts suggests that we observe them at a stage in their transformation intocompact spheroids, where a high density of dynamically unstable gas leads toefficient star formation and black hole accretion. For the one system where weprobe the mass of the gas reservoir, we find a gas fraction of just $0.06 \pm0.04$ and a depletion timescale of $50 \pm 40$ Myr, suggesting it istransitioning into quiescence. In general, we expect that the extreme level ofstar formation in the compact quasar host galaxies will rapidly exhaust theirgas reservoirs and could quench with or without help from active galacticnuclei feedback.