We propose an efficient and robust method to estimate the halo concentration based on the first moment of the density distribution, which is $R_1\equiv \int_0^{r_{\rm vir}}4\pi r^3\rho(r)dr/M_{\rm vir}/r_{\rm vir}$. We find that $R_1$ has a monotonic relation with the concentration parameter of the NFW profile, and that a cubic polynomial function can fit the relation with an error $\lesssim 3\%$. Tests on ideal NFW halos show that the conventional NFW profile fitting method and the $V_{\rm max}/V_{\rm vir}$ method produce biased halo concentration estimation by $\approx 10\%$ and $\approx 30\%$, respectively, for halos with 100 particles. In contrast, the systematic error for our $R_1$ method is smaller than $0.5\%$ even for halos containing only 100 particles. Convergence tests on realistic halos in $N$-body simulations show that the NFW profile fitting method underestimates the concentration parameter for halos with $\lesssim 300$ particles by $\gtrsim 20\%$, while the error for the $R_1$ method is $\lesssim 8\%$. We also show other applications of $R_1$, including estimating $V_{\rm max}$ and the Einasto concentration $c_{\rm e}\equiv r_{\rm vir}/r_{-2}$. The calculation of $R_1$ is efficient and robust, and we recommend including it as one of the halo properties in halo catalogs of cosmological simulations.

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.

We develop a method to identify proto-clusters based on dark matter halos at high redshift. Our main findings include:

• The test with N-body simulations shows that our finder has completeness $\sim 85\%$, purity $\geq 90\%$, mass estimates uncertainty $ \leq 0.25 {\rm dex}$.
• Our method can recover progenitor stellar mass distribution, providing an avenue to link high-z and low-z galaxies in clusters.

We extend the halo-based group finder to use data simultaneously with either photometric or spectroscopic redshifts. The performance is evaluated with a mock from N-body simulation. Our main results include:

• For magnitude $z \leq 21 $ galaxies in DESI, $\geq 60\%$ members in $\sim 90\%$ halos with $M_{\rm h} \geq 10^{12.5} h^{-1}M_\odot$ can be identified. Detected groups with $M_{\rm h} \geq 10^{12} h^{-1}M_\odot$ has purity $\geq 90\%$.
• Group mass assignment has uncertainty from 0.2 dex (high mass end) to 0.45 dex (low mass end).
• Group with 10 members has redshift accuracy $\sim 0.08$.
• A group catalog is provided for DR8.

We use a large N-body simulation to study the relation of structural properties of dark matter halos to their assembly history and environment. Our main conclusions are:

• The complexity of individual halo assembly histories can be well described by a small number of principal components, which are preferred over formation times for several reasons.
• 60%, 10%, 20% of the variances in halo concentration, axis ratio and spin, respectively, can be explained by combining four dominating predictors $\rm PC_{MAH,1}$, $M_{\rm halo}$, $\alpha_\mathcal{T}$, $b$. Degeneracies between predictors are found and analyzed, and are still hold for mass-binned samples.
• Tidal field provides important environmental information, with $\alpha_\mathcal{T}$ shows strongest assembly bias signal.

High-z spectroscopic surveys, usually incomplete in redshift sampling, present both opportunities and challenges to identifying groups in the high-z Universe. We develop a group finder that is based on incomplete redshift samples combined with photometric data. Our main findings are:

• Mock test shows that $\geq 90\%$ of groups with $M_{\rm h}\geq 10^{12} h^{-1}{\rm M}_\odot$ are successfully identified.
• The standard deviation in the halo mass estimation is smaller than 0.25 dex at all masses.
• We apply our group finder to zCOSMOS-bright and describe basic properties of the group catalog obtained.