The Sun is huge - its diameter is about 1.3 million km. For comparison, the Earth is still pretty big, but its diameter is only about 12,000 km.
The Sun is made up of plasma, which is a hot ionized gas. When physicists study the overall, large scale (macroscale) behavior of the Sun, or the overall behavior of large scale solar features like convection cells, prominences, sunspots, and coronal mass ejections, we treat the plasma like a fluid of charged particles that responds to electric and magnetic fields. We describe the large scale physics in terms of magnetohydrodynamic (MHD) equations that account for both the fluid and electromagnetic behavior of plasmas. MHD is part of what we would call classical physics.
When we want to discuss the microscopic processes like magnetic reconnection that occur in solar features like prominences, sunspots, etc., we use something called kinetic theory. Kinetic theory is an interesting tool because it helps us to get an idea of what individual electrons and ions in a plasma are doing without having to precisely track the individual motions of every particle in the plasma, since this would be extremely difficult. Instead, kinetic theory combines our understanding of how individual particles should move with the statistical, or average behavior of groups of particles using the Vlasov equation and the Boltzmann equation. Kinetic theory falls under the category of statistical physics, which uses tools of statistics and probability but still deals with larger scale sizes than quantum mechanics.
To get to quantum mechanics, we have to go down to the scales of the individual electrons and ions in the Sun. The nuclear reactions that fuse hydrogen into helium in the Sun's core would not work without quantum tunneling. In quantum mechanics, we describe the motion of individual particles using wave functions and probabilities described by the Shrodinger equation and Heisenberg uncertainty principle. The repulsive electrical force between the two positively charged particles is too great for even a single pair of protons in the Sun's core to overcome, but in quantum mechanics the weird behavior of individual particles on subatomic scales makes nuclear fusion possible. This article does a pretty good job of explaining how it works.
https://www.forbes.com/si...-shine/#3550f08c43f7 Quantum mechanics only is relevant to interactions on subatomic scales, so the entire Sun cannot "quantum tunnel" because it is much too large for quantum mechanical effects to apply. However, quantum tunneling is vital to the process of nuclear fusion that occurs deep within the Sun's core.