  We propose a novel method to quantify the assembly histories of dark matter
halos with the redshift evolution of the mass-weighted spatial variance of
their progenitor halos, i.e. the protohalo size history. We find that the
protohalo size history for each individual halo at z~0 can be described by a
double power-law function. The amplitude of the fitting function strongly
correlates to the central-to-total stellar mass ratios of descendant halos. The
variation of the amplitude of the protohalo size history can induce a strong
halo assembly bias effect for massive halos. This effect is detectable in
observation using the central-to-total stellar mass ratio as a proxy of the
protohalo size. The correlation to the descendant central-to-total stellar mass
ratio and the halo assembly bias effect seen in the protohalo size are much
stronger than that seen in the commonly adopted half-mass formation time
derived from the mass accretion history. This indicates that the information
loss caused by the compression of halo merger trees to mass accretion histories
can be captured by the protohalo size history. Protohalo size thus provides a
useful quantity to connect protoclusters across cosmic time and to link
protoclusters with their descendant clusters in observations.
