What is the origin of the Sun light? The Sunlight is an electromagnetic radiation.
Is it known what is the origin of this radiation? Can it be adequately described by classical electrodynamics (Maxwell's equations) as a motion of electric charges in the Sun? Is it necessary to take into account quantum effects described by quantum electrodynamics? Or is it necessary to take into account other processes?
 A: The light from the Sun comes from the photosphere; a relatively thin layer, a few hundred km thick.
The photosphere of the Sun is in radiative equilibrium, getting neither hotter or colder on average.
What this means is that the emission processes that produce the radiation that escapes from the photosphere, are the inverse of the absorption processes that stop radiation from deeper, hotter layers reaching us.
The dominant continuum process is bound-free photoionisation of H$^{-}$ ions that form when hydrogen atoms capture electrons released from the ionisation of potassium and sodium atoms in the atmosphere. There are some other bound-free photoionisation processes of other species that contribute continuum opacity, and bound-bound transitions between energy levels in a variety of atoms and ions that contribute opacities at discrete wavelengths.
The principle of detailed balance means that these absorption processes are balanced by free-bound photorecombination of H$^-$ ions contributing light over a continuum of wavelengths and bound-bound downward transitions in atoms and ions at specific wavelengths.
The understanding of these processes certainly requires quantum physics and cannot be described by classical electromagnetism.
A: The origin of sunlight is the hot plasma at and near its surface. It can be reasonably well described by Planck's black body radiation.
Check out https://en.m.wikipedia.org/wiki/Sunlight and sources therein.
A: Stars undergo nuclear fusion of chemical elements in their cores. The outward pressure of the resultant radiation and the thermal pressure from the plasma counter-act the impending collapse from the inward gravitational pressure of the star. Thus, a star is a delicate dance between outward and inward pressures. Eventually, the star runs out of nuclear fuel and the gravitational collapse wins over and the stars ends its life - depending on its mass and metallicity - as either a white dwarf star, a neutron star, or a black hole (or maybe even more exotic, undiscovered possibilities).
In terms of quantum mechanics there's a rich history of scientific progress about stellar models. For a nice historical review, I recommend this article by Arny. Therein, he discusses how purely classical models prior to the 1900's were used to try to estimate the age of the sun using thermodynamics. But this resulted in ~10^6 years, which they knew was incorrect due to the geological record. Some advancements were made in the thermodynamic understanding of plasmas by Jeans, Lambden and others.... But the real breakthrough came from quantum mechanics in two ways:

*

*spectroscopy allowed for the precise characterization of stars according to their composition - certain wavelengths of light are emitted by certain energy differences of the states of certain atoms. This paved the way to a nuclear theory of how stars work.


*Eddington and others computed how much energy would be required to keep the sun shining at its luminosity, and this led to numerous avenues of research. I won't go into detail here, the article by Arny is great. But the punchline is that the energy could be acquired through a nuclear process (such as the proton-proton chain reaction which requires 4 hydrogen atoms to produce helium isotope in low-mass stars such as our sun, or the CNO cycle which is thought to dominate in higher mass stars, and this is a field of its own since the late 1930's called stellar nucleosynthesis).

Eddington, in The Internal Constitution of the Stars (1926),
calculated without knowing how the energy was released, the energy yield of
hydrogen burning. He further noted that hydrogen was the most efficient fuel in
terms of energy per gram and that it would be able to power the sun for some 100
billion years.

But there were many details that were unexplained. Most importantly, how do they overcome the Coulomb barrier? The famous physicist George Gamow found the answer in quantum theory:

The key to the solution to this dilemma was found by G. Gamow in 1928 when he
showed that quantum mechanical tunneling allowed fusion to occur at temperatures
far lower than previously seemed plausible.

EDITED: Although fusion occurs in the core and produces photons as a by-product, these photons are rapidly absorbed by the plasma and re-emitted continually. It takes hundreds of thousands of years for photons to scatter through the opaque interior. Light leaving the surface of the star scatters from the photosphere of the stellar atmosphere, where the density of the plasma becomes sufficiently low for the photons to escape (a "last scattering surface"). Their path can be modeled as a random walk.
@my2cts points out that the spectrum of the photosphere is well-modeled by the black-body radiation spectrum.
EDIT: per the OP's question in the comments, as to whether a certain nuclear reaction occurs depends on what nuclear force mediates it (among other things...). This free article explains quite nicely in detail. For the simple example, consider the proton-proton reaction: 4 protons simultaneously colliding is extremely unlikely, so a series of 2-body interactions (chains) instead,
strong interaction: p + p $\rightarrow$ $^2$He, this does not work since $^2$He is highly unstable, i.e. $^2$He $\rightarrow$ 2p immediately (for the same reason p + $^4$He $\rightarrow$ $^5$Li is impossible).
weak interaction (Bethe 1938): p + p $\rightarrow$ D + e$^{+}$ + $\nu$, and this does work, where D is deuterium and $\nu$ is a neutrino.
