assembly bias signal (i.e. the difference between the red and blue lines) decreases with increasing redshift for halo samples selected by age. For concentration, the evolution of halo assembly bias is stronger in the sense that the clustering differences reverse at high redshift. These trends are in agreement with the evolution of the halo assembly bias signal found in the original Millennium simulation by [@Gao:2007] where they found the same trends when the peak height increases from 0.76 to 2.01, that is the increase of peak height from our samples between $z=0$ and $z=3$ (see @Wechsler:2006 for a comparison using the nonlinear mass for collapse.)

To understand the origin of this difference between using age and concentration as the secondary parameter we show in the right panel of Fig. \[Fig:CF\_Ev\] the CF of the main progenitors of the $z=0$ haloes selected at $z=1, 2$ and $3$. We call this sample the ‘tracking sample’. For this sample only, the secondary property halo labels (i.e. in terms of the extremes of concentration or formation time) refer to the $z=0$ descendants. We find different trends for different scales. At large scales ($>10\ h^{-1}{\rm Mpc}$), the tracking sample shows the same clustering trend as 