  We use hydrodynamics and radiative transfer simulations to study the 21~cm
signal around a bright QSO at $z \sim 10$. Due to its powerful UV and X-ray
radiation, the QSO quickly increases the extent of the fully ionized bubble
produced by the pre-existing stellar type sources, in addition to partially
ionize and heat the surrounding gas. As expected, a longer QSO lifetime,
$t_{\rm QSO}$, results in a 21~cm signal in emission located at increasingly
larger angular radii, $\theta$, and covering a wider range of $\theta$. Similar
features can be obtained with a higher galactic emissivity efficiency, $f_{\rm
UV}$, so that determining the origin of a large ionized bubble (i.e. QSO vs
stars) is not straightforward. Such degeneracy could be reduced by taking
advantage of the finite light traveltime effect, which is expected to affect an
HII region produced by a QSO differently from one created by stellar type
sources. From an observational point of view, we find that the 21 cm signal
around a QSO at various $t_{\rm QSO}$ could be detected by SKA1-low with a high
signal-noise ratio (S/N). As a reference, for $t_{\rm QSO} = 10\,\rm Myr$, a
S/N $\sim 8$ is expected assuming that no pre-heating of the IGM has taken
place due to high-$z$ energetic sources, while it can reach value above 10 in
case of pre-heating. Observations of the 21~cm signal from the environment of a
high-$z$ bright QSO could then be used to set constraints on its lifetime, as
well as to reduce the degeneracy between $f_{\rm UV}$ and $t_{\rm QSO}$.
