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ABSTRACT Recent observations from the EIGER JWST program have measured for the first time the quasar–galaxy cross-correlation function at $$z\approx 6$$. The autocorrelation function of faint $$z\approx 6$$ quasars was also recently estimated. These measurements provide key insights into the properties of quasars and galaxies at high redshift and their relation with the host dark matter haloes. In this work, we interpret these data building upon an empirical quasar population model that has been applied successfully to quasar clustering and demographic measurements at $$z\approx 2\!-\!4$$. We use a new, large-volume N-body simulation with more than a trillion particles, FLAMINGO-10k, to model quasars and galaxies simultaneously. We successfully reproduce observations of $$z\approx 6$$ quasars and galaxies (i.e. their clustering properties and luminosity functions), and infer key quantities such as their luminosity–halo mass relation, the mass function of their host haloes, and their duty cycle/occupation fraction. Our key findings are (i) quasars reside on average in $$\approx 10^{12.5}\, {\rm M}_{\odot }$$ haloes (corresponding to $$\approx 5\sigma$$ fluctuations in the initial conditions of the linear density field), but the distribution of host halo masses is quite broad; (ii) the duty cycle of (UV-bright) quasar activity is relatively low ($$\approx 1~{{\ \rm per\ cent}}$$); (iii) galaxies (that are bright in [O iii]) live in much smaller haloes ($$\approx 10^{10.9}\, {\rm M}_{\odot }$$) and have a larger duty cycle (occupation fraction) of $$\approx 13~{{\ \rm per\ cent}}$$. Finally, we focus on the inferred properties of quasars and present a homogeneous analysis of their evolution with redshift. The picture that emerges reveals a strong evolution of the host halo mass and duty cycle of quasars at $$z\approx 2\!-\!6$$, and calls for new investigations of the role of quasar activity across cosmic time.