  The Hubble constant ($H_0$) tension is one of the major open problems in
modern cosmology. This tension is the discrepancy, ranging from 4 to 6
$\sigma$, between the $H_0$ value estimated locally with the combination of
Supernovae Ia (SNe Ia) + Cepheids and the cosmological $H_0$ obtained through
the study of the Cosmic Microwave Background (CMB) radiation. The approaches
adopted in Dainotti et al. 2021 (ApJ) and Dainotti et al. 2022 (Galaxies) are
introduced. Through a binning division of the Pantheon sample of SNe Ia
(Scolnic et al. 2018), the value of $H_0$ has been estimated in each of the
redshift-ordered bins and fitted with a function lowering with the redshift.
The results show a decreasing trend of $H_0$ with redshift. If this is not due
to astrophysical biases or residual redshift evolution of the SNe Ia
parameters, it can be explained in light of modified gravity theories, e.g.,
the $f(R)$ scenarios. We also briefly describe the possible impact of high-$z$
probes on the Hubble constant tension, such as Gamma-ray bursts (GRBs) and
Quasars (QSOs), reported in Dainotti et al. 2022 (Galaxies) and Lenart et al.
2022 (ApJ), respectively.
