I have a thought experiment which in my understanding leads to faster than light messaging, and I had hoped the physics.SE community could help me resolve it. I should say that most of my understanding of quantum mechanics comes from these lectures by Feynman, although I am a mathematics major and can mostly see through the layman's filter he puts on. (He does that marvelously, by the way. I wish I was that good at it.)
Say you have a light source, two semi-transparent mirrors with 4% chance of reflecting a photon and one sensor. If you put the two mirrors parallel to eachother and shine light on it, then the sensor will have between $0\%$ and almost $16\%$ chance of registering a reflected photon, depending on the distance between the mirrors. Set the mirrors up for maximal constructive interference.
If you have full control of the timing of the light source (sending, say, one photon exactly once every millisecond), and the sensor is capable of differing by the time it takes the photon to be reflected from the first mirror and from the second mirror, then as I understand it the interference phenomenon would disappear. You will just get a (bit less than) $8\%$ chance of registering a photon.
Here's the problem: If I have another sensor on the other side of the mirrors, then it would, depending on the time resolution of the first sensor, either register $84\%$ or $92\%$ (i.e. the rest) of the photons sent by the source. If the first sensor is very far away from the mirrors and second sensor, then changing time resolusion (or just swapping the sensor) would have an immediate effect on the number of photons registered at the second sensor.
I don't believe this to be true, but as far as my understanding of quantum mechanics goes, that's the result I get.