  We derive new bounds on a heat current flowing into a quantum $L$-particle
system coupled with a Markovian environment. By assuming that a system
Hamiltonian and a system-environment interaction Hamiltonian are extensive in
$L$, we show that the absolute value of the heat current scales at most as
$\Theta (L^3)$ in a limit of large $L$. Also, we present an example that
saturates this bound in terms of scaling: non-interacting particles globally
coupled with a thermal bath. However, the construction of such system requires
many-body interactions induced by the environment, which may be difficult to
realize with the current technology. To consider more feasible cases, we focus
on a class of system where any non-diagonal elements of the noise operator
(derived from the system-environment interaction Hamiltonian) become zero in
the system energy basis, if the energy difference is beyond a certain value
$\Delta E$. Then, for $\Delta E = \Theta (L^0)$, we derive another scaling
bound $\Theta (L^2)$ on the absolute value of the heat current, and the
so-called superradiance belongs to a class to saturate this bound. Our results
are useful to evaluate the best achievable performance of quantum-enhanced
thermodynamic devices, which contain far-reaching applications for such as
quantum heat engines, quantum refrigerators and quantum batteries.
