Is there a simple layman way to explain the incompatibilities between quantum mechanics and (general) relativity to high school students? Is there a simple layman way that I can use to explain the incompatibilities between quantum mechanics and (general) relativity to high school students (people with not much knowledge of the intricate math of quantum mechanics and (general) relativity)?
 A: For what it's worth, there's a simple argument which explains the need for quantum gravity, using just dimensional analysis:


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*Quantum mechanics attaches a length scale $l$ to every mass $m$, called the Compton wavelength $l \sim \frac{h}{m c}$. If you consider a massive object (particle), at distances comparable to this (and smaller), quantum effects become strong.

*General relativity attaches a length scale $l$ to every mass $m$, called the Schwarschild radius $l \sim \frac{G M}{c^2}$. If you consider a massive object, at distances comparable to this (and smaller), general relativistic effects become strong.
Equating the two, we can derive a special scale called the Planck scale. An imaginary particle with Planck mass has a Compton wavelength and Schwarschild radius of about the same size, so for such particles (i.e. when we deal with such energy scales) both general relativistic effects and quantum effects become strong -- this is why we really need a theory incorporating both.
As for why combining the two is hard:


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*GR tries to use physics to describe the geometry of spacetime. Due to quantum effects, there will be (severe) "quantum fluctuations"  in the geometry of spacetime! So, in a sense, the problem is that we have no simple solution which we can use as a crutch. In physics, we almost always solve a simple case and perturb around that solution to push as far as possible. If perturbation theory fails (as it does for GR+QM) we're at a loss for what to do.

*From the perspective of particle physics, if you want to "zoom in" and probe what happens at short distances, then you use very energetic particles whose Compton wavelength is comparable to your length scale. However, as you keep increasing the energy of your particles, at the Planck mass, their Schwarschild radius overtakes the Compton wavelength. So even though your particles are very energetic, they form blackholes and stop you from probing small distances!
A: Probably no simple explanation. It is however important to emphasize that the incompatibility applies only to general relativity. The special relativity and quantum mechanics are very compatible and were luckily married many decades ago, giving birth to the quantum field theory which is an incredibly successful framework in which physicists built the quantum electrodynamics, quantum flavordynamics, quantum chromodynamics and the whole standard model. The whole modern quantum physics would not be thinkable without combining quantum mechanics with special relativity.
General relativity is a different case however. The root cause of the issue is rather technical, so laymen terms do not reasonably work here. Basically, when you try to quantize gravity, you get nonsensical (infinite) results that cannot be remedied. A solution to this problem is not yet known.
