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We first present new Subaru narrow-band observations of the Ly$\alpha$ haloaround the quasi-stellar object (QSO) CFHQ J232908$-$030158 at $z=6.42$, whichappears the most luminous and extended halo at $z>5$($L_{Ly\alpha}=9.8\times10^{43}$ erg s$^{-1}$ within $37$ pkpc diameter). Then,combining these measurements with available data in the literature, we find twodifferent evolutions of QSOs' Ly$\alpha$ halos. First is a possible short-termevolution with QSO age seen in four $z>6$ QSOs. We find the anti-correlationbetween the Ly$\alpha$ halo scales with QSOs' IR luminosity, with J2329-0301'shalo being the brightest and largest. It indicates that ionizing photons escapemore easily out to circum-galactic regions when host galaxies are less dusty.We also find a positive correlation between IR luminosity and black hole mass($M_\text{BH}$). Given $M_\text{BH}$ as an indicator of QSO age, we propose ahypothesis that a large Ly$\alpha$ halo mainly exists around QSOs in the youngphase of their activity due to a small amount of dust. The second is anevolution with cosmic time seen over $z\sim2-5$. We find the increase ofsurface brightness toward lower-redshift with a similar growth rate to that ofdark matter halos (DHs) which evolve to $M_\text{DH}=10^{12}-10^{13}$ M$_\odot$at $z=2$. The extent of Ly$\alpha$ halos is also found to increase at a ratescaling with the virial radius of growing DHs, $r_\text{vir} \proptoM_\text{DH}^{1/3}$($1+z$)$^{-1}$. These increases are consistent with ascenario that the CGM around QSOs evolves in mass and size keeping pace withhosting DHs.

Possible evolution of the circum-galactic medium around QSOs with QSO age and cosmic time revealed by Ly$\alpha$ halos