How do quantum mechanics and general relativity conflict?
One example is quantum entanglement.
Off the top of my head, one conflict is between quantum entanglement and relativity, though this conflict has been "explained away" in a sense.
Two objects are entangled if knowing information about one object allows you to know information about a second object, which you have not made a direct measurement of.
Quantum mechanics says that if the state of an object/particle is unknown, before a measurement is made, the object simultaneously possesses all possible values for that measurement. When a measurement is made, the superposition state is disturbed, the wave function collapses, and a single value/property is observed.
Quantum entanglement says that if two objects are entangled and a measurement of the state of one of the objects is made, the wave function of the first object collapses from the superposition of states to a single possibility, which causes the wave function of the entangled object to collapse as well, giving both objects "definite" properties when only one of the objects was measured. This implies that the objects are "communicating" in a sense.
Einstein tried to use this notion to disprove QM by devising a thought experiment which, in essense, said that if you took two entangled objects to opposite sides of the universe and made a measurement of one, somehow that object would have to send information to the other object and tell it which state it needed to assume. If you had your friend on the opposite side of the universe make a measurement of one of the entangled objects a split second after you did, that information would have to travel an unfathomably large distance in a very small amount of time, i.e. faster than the speed of light (EPR paradox). This directly violates Einstein's theory of relativity.
However, the no-communication theorem proposed that QM does not actually result in a paradox, and relativity is not violated there.
This is just one example; there are other reasons why QM and relativity do not get along so well.