So we have two more cases to consider:
Case 3: we pulse the laser with very short pulses, emitting only one photon at a time. This is actually not possible with a laser, but it is possible with something like this single-photon-emitting light source (which was actually case 5 when I first made up the list).
Before analyzing this case I have to hedge: I'm pretty sure I know the right answer, but not 100%. I have actually asked one of the authors of that paper to confirm my suspicions but he's busy and so it will be a while before I expect to get an answer. If his answer turns out to be substantially different from what I say here I will definitely let you all know.
So with that disclaimer in mind, I'm quite confident that this case will turn out to have the same behavior as the case where we have a laser dimmed by a filter, and the photon emissions post-filter detected using a parametric down-converter, i.e. there will be no (first-order) interference (with one exception, which will arrive at in case 4). The reason I'm confident is that this case is structurally the same as that one: we have made a modification to the experiment that allows us to know when the photon was emitted, which allows us to determine which path the photon took by comparing departure and arrival times, and so there can be no interference. Please note that I have taken pains not to say that our (potential) knowledge of the emission time causes the interference to go away. It doesn't. What causes the loss of interference is the entanglement of the photon with something else. Entanglement is a pre-requisite for measurement, which is a pre-requisite for knowledge, so it is true that if there is (potential) knowledge that there can be no interference, but the knowledge bit is a red herring. The only reason I'm using that phraseology is that it is, sadly, ubiquitous in QM pedagogy.
So, with that out of the way, let us proceed to the interesting case: We turn the laser on and off with a duty cycle of 50% and a period that is long enough that the pulses actually do produce interference. The lengths of the arms of the interferometer are adjusted so that the Nth pulse coming from the long arm exactly coincides at the detector with the N+Kth pulse coming from the short arm for some integer K>=1.
Before I go into detail on this I want to say a word about how practical such an experiment might be. To make this work, the period of the pulses has to be long enough that the individual pulses are coherent, but short enough that we can "store" at least one them "in flight" while waiting for the next one without losing coherence. Can this actually be done? Yes, it can, at least if you believe Wikipedia. There is says that fiber lasers can be built with coherence lengths exceeding 100km. That's about 300 microseconds, which is essentially forever by the standards of quantum optics. You could actually do this experiment with a regular semiconductor laser with a coherence length of "merely" 100m. It's pretty straightforward to power-cycle a laser, produce time delays, and measure the results with events happening at nanosecond/meter scales. So this would not even push the boundaries of the state of the art.
There is no doubt what would happen in this case: you would see interference, but (and this is really important) only after the Kth cycle. Before that we know that all of the light at the detector arrived via the short arm, so there is no interference. After the Kth cycle, light is arriving from both arms, so we do get interference. There would be some transient effects at the beginning and end of each pulse, but at steady-state the interference would be easily detectable. This is predicted both by quantum theory and classical E-M theory. There is absolutely nothing interesting going on here, until, that is, you make one more little modification: in addition to a 50% duty cycle, you also dim the laser with a filter.
The outcome predicted by QM is clear: dimming the laser with a filter makes no difference. Whatever we saw when the laser was bright we should also see when the laser is dim, namely, for the first K cycles of the laser we get no interference.
How is this possible? The dramatic narrative of interference usually goes something like this: interference happens when a photon (or some other particle, it doesn't really matter) is placed in a quantum superposition, usually a quantum superposition of physical locations. The usual slogan is "the photon goes both ways". The two paths are then brought together in such a way that no which-way information is available in the final state. The result is interference.
But can this possibly be happening in this case? The two paths that the photon can take are separated by an enormous amount of time, big enough that we are able to turn the laser off and back on while we are waiting for it to travel the long way 'round through our interferometer. It's already enough of a mind bender to say that we cannot know when a particular photon was emitted when the laser is on continuously, but now we seem to be going a step further: in order to have interference, we cannot even know which power cycle a detected photon was produced by! Is it really possible for a laser to produce a photon that is in a quantum superposition across power cycles? That seems extremely improbable. Surely once you turn the laser off, the universe is committed: there's a batch of photons flying through space at the speed of light in some particular quantum mechanical configuration. Surely that configuration can't be changed by something that happens (or not -- we could choose at any time not to turn the laser back on!) in the future?
There is another possibility. Maybe the interference we see is not created by one photon interfering with itself, but rather interfering with another photon produced in a different cycle. This seems a lot more plausible, but is it actually possible? Paul Dirac, one of the founders of quantum theory, once famously wrote “...each photon then interferes only with itself. Interference between different photons never occurs.” Interestingly, while I can find this quote all over the internet, and it is invariably attributed to Dirac, I cannot locate its original source. So it's possible that this quote is apocryphal. But it doesn't matter. What matters is that this sentiment was taken seriously for decades until an experiment by Magyar and Mandel debunked it in 1963. There have been books written about, and even entitled, multi-photon interference. So it's definitely a thing.
The experiment as we have described it to this point is an interesting variant on the Magyar and Mandel experiment. There they used two different lasers (actually they were masers, but it doesn't matter) to generate their photons, whereas we are using one laser and separating the production of the two photons by time using power cycles and bringing them back together using delay lines. But it amounts to the same thing. The key is that we're bring the photons back together at the same time. That's the reason that the delay time in the interferometer is an integer multiple of the cycle time on the laser, otherwise it doesn't work.
So this might be a mildly interesting but not earth-shattering result. Maybe someone has even done it, I don't know.
But there is one thing that should make us a little queasy about this line of thought, and that is that we cannot actually control how many photons enter the interferometer on any given cycle. We can attenuate the beam so that on average we get one photon per cycle, but that will only be an average. Some cycles we will get more than one, some cycles we will get none. If we're depending on photons to interfere with each other then we need the same number of photons each cycle, otherwise some photons won't have partners to interfere with.
In fact, we can actually completely eliminate the possibility that what we see is multi-photon interference simply by making the laser even dimmer. Let's attenuate the laser to the point where, rather than of one photon every cycle, we instead get one photon every 2K cycles (or more). In other words, most of the time the interferometer is totally dark. Every now and then we will get a photon. The temporal separation between photons is now much more than the coherence time of the laser, much more than the cycle time of the laser, much more than the travel time through the long arm of the interferometer. We can make it an hour or more between photons. Theory predicts that we will still see interference! Not only that, but it will be the exact same interference pattern that we saw when the interferometer was fully illuminated.
Note well that this is in no way intended to highlight a problem with QM. The outcome predicted by QM is very clear, and I would give you very long odds that this prediction is correct. The problem is only in trying to tell a story about what the fleep is going on here that involves photons being emitted by the laser. I don't see any way to do it.
In fact, I'll go one step further: AFAICT, this is a strong argument for the following remarkable conclusion (and if this holds up I think it really could be earth-shattering): the quantum wave function must be physically real because that is the only thing that could account for the K-cycle delay in the onset of interference. If anyone can see a flaw in my reasoning I would really appreciate it if you would point it out.