  Quasars at $z \,=\, 6$ are powered by accretion onto supermassive black holes
with masses $M_{\rm BH} \sim 10^9 \rm \, M_{\odot}$. Their rapid assembly
requires efficient gas inflow into the galactic nucleus, sustaining black hole
accretion at a rate close to the Eddington limit, but also high central star
formation rates. Using a set of cosmological 'zoom-in' hydrodynamic simulations
performed with the moving mesh code Arepo, we show that $z \,=\, 6$ quasar host
galaxies develop extremely tightly bound stellar bulges with peak circular
velocities $300$ - $500$ km s$^{-1}$ and half-mass radii $\approx 0.5 \, \rm
kpc$. Despite their high binding energy, we find that these compact bulges
expand at $z \, < \, 6$, with their half-mass radii reaching $ \approx 5$ kpc
by $z \, = \, 3$. The circular velocity drops by factors $\approx 2$ from their
initial values to $200$ - $300$ km s$^{-1}$ at $z \, \approx \, 3$ and the
stellar profile undergoes a cusp-core transformation. By tracking individual
stellar populations, we find that the gradual expansion of the stellar
component is mainly driven by fluctuations in the gravitational potential
induced by bursty AGN feedback. We also find that galaxy size growth and the
development of a cored stellar profile does not occur if AGN feedback is
ineffective. Our findings suggest that AGN-driven outflows may have profound
implications for the internal structure of massive galaxies, possibly
accounting for their size growth, the formation of cored ellipticals as well as
for the saturation of the $M_{\rm BH}$ - $\sigma_{\star}$ seen at high velocity
dispersions $\sigma_{\star}$.
