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Understanding the nature of high-$z$ dusty galaxies requires a comprehensiveview of their ISM and molecular complexity. However, the molecular ISM athigh-$z$ is commonly studied using only a few species beyond CO, limiting ourunderstanding. In this paper, we present the results of deep 3 mm spectral linesurveys using the NOEMA targeting two lensed dusty galaxies: APM 08279+5255(APM), a quasar at redshift $z=3.911$, and NCv1.143 (NC), a $z=3.565$ starburstgalaxy. The spectral line surveys cover rest-frame frequencies from about330-550 GHz. We report the detection of 38 and 25 emission lines in APM and NC,respectively. The spectra reveal the chemical richness and the complexity ofthe physical properties of the ISM. By comparing the spectra of the two sourcesand combining the gas excitation analysis, we find that the physical propertiesand the chemical imprints of the ISM are different between them: the moleculargas is more excited in APM, exhibiting higher molecular-gas temperatures anddensities compared to NC; the chemical abundances in APM are akin to the valuesof local AGN, showing boosted relative abundances of the dense gas tracers thatmight be related to high-temperature chemistry and/or XDRs, while NC moreclosely resembles local starburst galaxies. The most significant differencesare found in H2O, where the 448GHz H2O line is significantly brighter in APM,likely linked to the intense far-infrared radiation from the dust powered byAGN. Our astrochemical model suggests that at such high column densities, FUVradiation is less important in regulating the ISM, while CRs (X-rays/shocks)are the key players in shaping the abundance of the molecules and the initialconditions of star formation. Such deep spectral line surveys open a new windowto study the physical and chemical properties of the ISM and the radiationfield of galaxies in the early Universe. (abridged)

SUNRISE: The rich molecular inventory of high-redshift dusty galaxies revealed by broadband spectral line surveys