I know that the philosopher, J.A. Barrett, for example, has argued that quantum mechanics and special relativity are logically incompatible. Let me quote him:
"The Lorentz transformations and the principle of relativity directly entail something else [besides that c limits the speed of material objects] concerning the temporal relationships between events that is perhaps less familiar: Space-like separated events (events that cannot be signaled between luminally) have no canonical temporal order. More specifically, if events A and B are space-like separated, then there will be an inertial frame where an observer would take A to occur first and another inertial frame where an observer would take B to occur first. And from this it follows, by the principle of relativity, that there can be no physical matter of fact concerning the temporal order of the two events. That special relativity predicts that there is generally no physical matter of fact concerning the temporal order of events provides a straightforward way of showing the incompatibility of the standard formulation of quantum mechanics and relativity."
He then goes on with an example of how QM and SR can be shown to be incompatible with regard to the temporal order of events. His example deals with putting an electron in superposition in two boxes having synchronized clocks and with each having an alarm attached to the box. One (box E) has an alarm set to go off at noon on Jan 1, 2050 and the other (box F) at one minute past noon (on the same date). By taking box F far away from earth (more than one light-minute), there is a connundrum introduced when the alarm rings and both parties look inside the box. Here is what he says:
"Suppose that the initial state of the electron before either observer makes a measurement to locate it (in any inertial frame!) is
1/(sqrt(2))(|Earth>(sub)e- + |Far Away>(sub)e-)
Now suppose that the standard formulation of quantum mechanics is right and that the collapse dynamics describes the time-evolution of quantum mechanical states whenever there is a measurement interaction. What is the physical state of the electron just before [box F person's] measurement? Since the measurement events are space-like separated, there will be an inertial frame E where [box E person's] measurement [is] first. In this case, [box E person's] measurement caused a collapse, and the state just before [box F person's] measurement is either (1) |Earth>(sub)e- or (2) |Far Away>(sub)e- with probability 1/2 in each case. But since the measurement events are space-like separated, there will also be an inertial frame F where [box F person] is the first to look for the electron. In this case the state of the electron just before [box F person's] is
(3) 1/(sqrt(2))(|Earth>(sub)e- + |Far Away>(sub)e-)
[S]ince [box E person] has not yet interacted with the electron. It should be noted that states (1) and (2) are mutually exclusive and that each is flatly incompatible with state (3). On the standard interpretation of quantum mechanical states, state (1) describes an electron that is determinitely in the box on Earth, state (2) describes an electron that is determinately in the far-away box, and state (3) describes an electron that has no determinate position whatsoever. Indeed, quantum mechanics entails that there are experiments that would empirically distinguish between (1), or (2) and state (3). Since the standard formulation of quantum mechanics requires mutually incompatible states for different inertial frames, it is flatly inconsistent with the principle of relativity. So quantum mechanics and special relativity are in this sense logically incompatible." [Barrett, J.A., "Are our Best Physical Theories (Probably and/or Approximately) True", 2002].
There is also possible incompatible issues dealing with violation of 'spooky' action at a distance which quantum violations of Bell's inequalities seem to imply. Of course, this issue is very controversial.
Warm regards, Harv