Is Quantum Tunneling necessary to explain why The Sun keeps shining , due to it not being hot enough to provide the necessary energy for protons to collide?
There is no classical explanation on how is shining and has been shining for 4.6 billion years. The only way to explain that is through quantum mechanics.
Well, there's no *successful* explanation. The best one I know of is that it's a giant ball of smoldering coal. It gets the temperature and appearance right (assuming there's a lot of oxygen available) but only allows it to burn for a few thousand years.