Really delayed choice quantum eraser

[mgiddy911]

I am a High School Senior in Advanced Physics Topics class. We have recently deiscussed the basic double slit experiments and ways of quntum erasing and delaying choices, and how those situation affect the outcome of the screen whether it will show an interference pattern or not.
We then discussed a situation for delayed chice quantum erasing where you have a photon source that is split by a beam splitter, so now there is two possible paths. On each path there is a down converter which sends a signal photon towards a screen and and idle photon towards another splitter. The signal photons go to the screen and the dile photons hit the splitter and have two possible paths each. one path of each idler photon goes to a detector call them (1 and 4), the other path of each photn leads it to one final splitter with each of its possible paths going to a detector( 2 and 3).
the principal is that there's a 50 50 chance that the idler photons will be detected by either detector 1 or 4, if that happens we have which path information and we would not see an interference pattern, but if they are detected by 2 or 3 then we would not have which path information and there would be an interference pattern.
So if you let 100 photons go from the source, then you would just see a jumble on the screen until you took the data from the detectors and figured out which photons were deteted, then you could map them out on the screen and see some formed the interference patttern and others just made their clumps from being detected.
What i was wondering (sorry this took so long) was what would happen if you made the path of the idler photons between the down converter and their beam splitters immensely long, for example from the down converter to the splitter was 99999999 lightyears. Then the signal photons would hit the screen before the idler photon would have a chance to be detected. If you observed the screen before the idler photon could be detected what would you see?
Another idea I had that might change the outcome is what if you built the downconverter so that i received a photon and emitted radioactive nucleii, so that one would go to the screen and the other would go to the splitter. Then there is even less of a chance that the idler nucleii would make it to the detector because they could decay before they got there. So if you had them traveling a long distance again like 999999 lightyears, you could again see the signal nucleii hit the screen before their idler nucleii had the chance to make it to the detector, but this time there is even less of a chance that they could make it to a detector. So how would this set up change the outcome of the screen?
Thank you in advance for all your help, I hope you can understand my post and sorry if it seems rushed but I need to get to class soon

 

[mgiddy911]

Here is a picture that may or maynot clarify the scenario, I drew it up real quick in Photoshoip sorry for its roughness.
What I am really trying to understand here is:
What would be seen on the screen if you have the distance for the idler photon from the down converters to their beamsplitters where they go to one of the detectors be incredibly large. If you did this unless there is already a flaw in my logic, you would see the signal photons hit the screen before the idler photons had a chance to be observed by the detectors which may or maynot provide whichpath info, (only would provide which path info from detectors 1 or 4). I think you would see an interference pattern on the screen since no observations were made, but does the potential for whichpath cause no interference pattern to form? If it does doens't this seem like predestination of some sorts

 

[vanesch]

I am a High School Senior in Advanced Physics Topics class.

You're doing delayed choice quantum erasers in High School ?

I could try to explain how it happens, but I need to use a bit the formalism of quantum mechanics for that - quantum erasers are tricky things! So I don't know if you are aware of this formalism (you know, ket vectors, projection postulate or Born rule and so on). And also some (not much) optics, essentially the Mach-Zehnder interferrometer.