Why can the following nuclear reaction happen in big stars but not on Earth? $$
\rm
{}^{12}C +{}^{12}C\rightarrow  {}^{20}Ne + {}^4He
$$
Why doesn't this reaction occur on Earth but can occur in big stars?
 A: The (nuclear) reaction you quote is highly exothermic, but that in itself is not enough for the reaction to proceed. It needs to be initiated and this requires certain physical conditions to be met.
Most importantly, the carbon nuclei need to be brought close enough together that the strong nuclear force can fuse them together. The strong nuclear force is short range, only acting over a few $10^{-15}$ m, so the nuclei need to achieve these separations. However, their positive charges provide a mutually repulsive force that prevents this even in the centre of a massive star.
A quantum effect called "tunneling" is required that provides a small probability $p$ that the carbon nuclei can get close enough to fuse, even if classically they don't have sufficient kinetic energy to be squashed close enough to fuse.
$p$ is enhanced by starting off with high kinetic energy, that lets the nuclei get moderately close together to allow tunneling to work, and this is how the fusion occurs in stars. The high interior temperatures give the carbon nuclei lots of kinetic energy. Such temperatures $(\sim 10^{9}$ K) do not occur naturally on Earth, although carbon nuclei can be (and have been) propelled towards each other with MeV energies in order to make the reaction occur.
It is also possible to initiate the reaction by crushing the nuclei close together with an external force like gravity. However, the densities required are of order $10^{14}$ kg/m$^3$, and these conditions are not found on Earth either.
A: The reaction has a positive $Q$-value but can only occur if two carbon-$12$ nuclei are close enough together.  
To get the nuclei in that position the coulomb repulsion between the nuclei must be overcome ie the nuclei must gain electric potential energy.  
This means that the nuclei must have sufficient kinetic energy when they are far apart so that the kinetic energy of the nuclei is converted into electric potential energy as the nuclei come closer together.  
For the nuclei to have sufficient kinetic energy there must be a very high temperature which only occurs in the centre of big stars.
