This post is (part of) the answer to a puzzle I posed here. Read that first if you haven't already.
To make this discussion concrete, let's call the time it takes for light to traverse the short (or Small) arm of the interferometer Ts, the long (or Big) arm Tb (because Tl looks too much like T1).
So there are five interesting cases here. Let's start with the easy one: we illuminate the interferometer with a laser which we turn on and leave on. In that case it's a no-brainer: the photons arrive at the business end of the interferometer first from the short path Ts seconds after turning the laser on. At this point we know (because timing) which way the photons went so there is no interference. Then at time Tb the photons arrive from the long path. All the photons are identical, so we no longer know which path they took. So interference.
Case 2 (the one the original puzzle was about): the laser is turned on and stays on, but the power is modulated (or filters are put in place) so that the light level is very low, so low that the average time of arrival between individual photons at the detector is much larger than Tb.
There are two plausible-sounding answers in this case. Plausible-sounding answer #1 is that the result is exactly the same as before: after Tb we still get interference. The equations of quantum mechanics are independent of brightness, so wherever we get interference when the light is bright, we still get interference when the light is dim.
Plausible-sounding answer #2 is that when the light is dim we can tell which way the photon went by looking at the timing. Whenever we get a detection there are two possibilities: either the photon was emitted Ts seconds earlier and took the short path, or the photon was emitted Tb seconds earlier and took the long path. So if we can tell when the photon was emitted, then there will be no interference.
But can we tell? Well (and this is the answer I was originally looking for when I composed the puzzle) it depends on exactly how we make the laser dim! There are at least two ways, and they produce different results.
The first is to put some sort of shutter in front of the laser that only lets through one photon through at a time. This is equivalent to turning the laser on only for very short periods of time. If we do it this way we will get no interference.
The other way is to do nothing to the laser itself, but rather to put a filter between the laser and the interferometer that blocks (or reflects) most of the photons. The photons are blocked by the filter at random, so there is no way to tell when a particular photon got through the filter. Hence: interference.
But suppose we tweak our setup slightly so that we can tell when a photon was transmitted by the filter. How can we do this? It is this exploration that leads to (IMHO) a profound insight.
Think about it: how do you detect a photon without destroying it? You can't! The only way to detect a photon is to have some atom absorb it, and that process destroys the photon. But there is a sneaky trick we can do: we can run the photons through a parametric down-converter (PDC). A PDC is a crystal made of some material (typically some stuff called beta-barium borate or BBO) whose atoms absorb photons at one wavelength and then re-emit them as two photons at different wavelengths. The key is that these two emissions happen at more or less the same time. So we can send one of these photons into the interferometer and use the other one to tell us when this event happened. Experimental physicists actually do experiments like this routinely. To distinguish between the two photons, the one that goes into the apparatus is called the signal photon, while the other that is measured to figure out the timing is called the idler. By measuring (and hence destroying) the idler photon we can tell when the signal photon entered the interferometer, and so we can tell which way the signal photon went (by comparing the timing of entry and exit). So we cannot have interference.
Here is the profound insight: this setup will not produce interference even if we don't actually measure the idler photon! Why? (You might want to think about that for a moment before proceeding.) Because if it did, then we could use that fact to transmit information faster than light!
Here is how we would do it: instead of measuring the idlers, we send them off (via mirrors) to some distant location (let's call it L1) At the same time, we take our interferometer and move it away from the PDC by the same distance but in the opposite direction to a location we will call L2. The distance between L1 and L2 is much greater than the length of the long arm of the interferometer.
If we could produce interference by choosing not to perform any measurements on the idlers then we could use this setup to communicate faster than light by selectively measuring the idlers or not. When we measured the idlers at L1, the interference would be destroyed at L2. And this effect would have to happen instantaneously because if it didn't then we could measure idlers at L1 and still have interference at L2, and that is impossible.
The profound conclusion is that photons emitted by a parametric down-converter do not produce interference! 
Those of you who have read my paper on the EPRG paradox will find this all to be familiar territory. In fact, this is the exact same conclusion that was reached in that paper, and for the exact same reason: the photons emitted by a PDC are entangled, and entangled photons do not self-interfere. The reason they don't self-interfere is that entanglement is the first step of the measurement process, and it, not measurement per se, is what destroys (first-order) interference.
This is all old news (at least 17 years old). So why is this (IMHO) cool? Because we could reach this conclusion without knowing anything about entanglement! We didn't need to invoke EPR or Bell's theorem or polarization or anything like that. All we needed was the principle that which-way information destroys interference to reach the conclusion that if there is any physical process that reliably produces multiple photons at the same time, then those photons cannot self-interfere. We have shown, without doing any math, only from elementary first principles, that entangled particles are different in some deep and profound way from non-entangled ones.
I think that's cool.
That's probably enough for one post. I'll finish up the other three cases later.
 This is not quite true. The strictly correct statement is that entangled photons do not produce first-order interference. They can and do produce second-order interference, which can only be detected by transmitting classical information from L1 to L2.