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 $sigma_{star}$.
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