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We present a multi-line survey of the interstellar medium (ISM) in two $z>6$quasar (QSO) host galaxies, PJ231-20 ($z=6.59$) and PJ308-21 ($z=6.23$), andtheir two companion galaxies. Observations were carried out using the AtacamaLarge (sub-)Millimeter Array (ALMA). We targeted eleven transitions includingatomic fine structure lines (FSLs) and molecular lines: [NII]$_{\rm 205\mu m}$,[CI]$_{\rm 369\mu m}$, CO ($J_{\rm up} = 7, 10, 15, 16$), H$_2$O$3_{12}-2_{21}$, $3_{21}-3_{12}$, $3_{03}-2_{12}$, and the OH$_{\rm 163\mu m}$doublet. The underlying far-infrared (FIR) continuum samples the Rayleigh-Jeanstail of the respective dust emission. By combining this information with ourearlier ALMA [CII]$_{\rm 158\mu m}$ observations, we explore the effects ofstar formation and black hole feedback on the galaxies' ISM using the CLOUDYradiative transfer models. We estimate dust masses, spectral indexes, IRluminosities, and star-formation rates from the FIR continuum. The analysis ofthe FSLs indicates that the [CII]$_{\rm 158\mu m}$ and [CI]$_{\rm 369\mu m}$emission arises predominantly from the neutral medium in photodissociationregions (PDRs). We find that line deficits are in agreement with those of localluminous infrared galaxies. The CO spectral line energy distributions (SLEDs),reveal significant high-$J$ CO excitation in both quasar hosts. Our CO SLEDmodeling of the quasar PJ231-20 shows that PDRs dominate the molecular mass andCO luminosities for $J_{\rm up}\le 7$, while the $J_{\rm up}\ge10$ CO emissionis likely driven by X-ray dissociation regions produced by the active galacticnucleus (AGN) at the very center of the quasar host [abridged].

An ALMA multi-line survey of the ISM in two quasar host-companion galaxy pairs at $z > 6$