  Galaxies occupy different regions of the
[OIII]$\lambda5007$/H$\beta$-versus-[NII]$\lambda6584$/H$\alpha$ emission-line
ratio diagram in the distant and local Universe. We investigate the origin of
this intriguing result by modelling self-consistently, for the first time,
nebular emission from young stars, accreting black holes (BHs) and older,
post-asymptotic-giant-branch (post-AGB) stellar populations in galaxy formation
simulations in a full cosmological context. In post-processing, we couple
new-generation nebular-emission models with high-resolution, cosmological
zoom-in simulations of massive galaxies to explore which galaxy physical
properties drive the cosmic evolution of the optical-line ratios
[OIII]$\lambda5007$/H$\beta$, [NII]$\lambda6584$/H$\alpha$,
[SII]$\lambda\lambda6717,6731$/H$\alpha$ and [OI]$\lambda6300$/H$\alpha$. The
line ratios of simulated galaxies agree well with observations of both
star-forming and active local SDSS galaxies. Towards higher redshifts, at fixed
galaxy stellar mass, the average [OIII]/H$\beta$ increases and [NII]/H$\alpha$,
[SII]/H$\alpha$ and [OI]/H$\alpha$ decrease -- widely consistent with
observations. At fixed stellar mass, we identify star formation history, which
controls nebular emission from young stars via the ionization parameter, as the
primary driver of the cosmic evolution of [OIII]/H$\beta$ and [NII]/H$\alpha$.
For [SII]/H$\alpha$ and [OI]/H$\alpha$, this applies only to redshifts above
$z=1.5$, the evolution at lower redshift being driven in roughly equal parts by
nebular emission from AGN and post-AGB stars. Instead, changes in the hardness
of ionizing radiation, ionized-gas density, the prevalence of BH accretion
relative to star formation and the dust-to-metal mass ratio (whose impact on
the gas-phase N/O ratio we model at fixed O/H) play at most a minor role in the
cosmic evolution of simulated galaxy line ratios.
