  We investigate cosmological constraints on the holographic dark energy (HDE)
using the state-of-the-art cosmological datasets: Planck CMB angular power
spectra and weak lensing power spectra, Atacama Cosmology Telescope (ACT)
temperature power spectra, baryon acoustic oscillation (BAO) and redshift-space
distortion (RSD) measurements from six-degree-field galaxy survey and Sloan
Digital Sky Survey (DR12 & DR16) and the Cepheids-Supernovae measurement from
SH0ES team (R22). We also examine the HDE model and $\Lambda$CDM with and
without $N_{\rm eff}$ (effective number of relativistic species) being treated
as a free parameter. We find that the HDE model can relieve the tensions of
$H_0$ and $S_8$ to certain degrees. With ``Planck+ACT+BAO+RSD'' datasets, the
constraints are $H_0 = 69.70 \pm 1.39\ \mathrm{km\ s^{-1} Mpc^{-1}}$ and $S_8 =
0.823 \pm 0.011$ in HDE model, which brings down the Hubble tension down to
$1.92\sigma$ confidence level (C.L.) and the $S_8$ tension to $1$-$2\sigma$
C.L. By adding the R22 data, their values are improved as $H_0 = 71.86 \pm 0.93
\,\mathrm{km\ s^{-1} Mpc^{-1}}$ and $S_8 = 0.813 \pm 0.010$, which further
brings the Hubble tension down to $0.85\sigma$ C.L. and relieves the $S_{8}$
tension. We also quantify the goodness-of-fit of different models with Akaike
information criterion (AIC) and Bayesian information criterion (BIC), and find
that the HDE agrees with the observational data better than the $\Lambda$CDM
and other extended models (treating $N_{\rm eff}$ as free for fitting).
