That's playing word games. Let's remove the word "measure" and insert the word "bloop". If you bloop something twice, you get the same result twice. Even in QM. And bubble chambers bloop a trajectory.
> If you think you can, describe the experimental setup for me.
It's called the Stern-Gerlach experiment. See the section "Sequential experiments" on wikipedia:
https://en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experime...
> But in order to see the vase, your eyes have to accumulate a lot of photons, and that takes time. So the vase on the table has to persist at least long enough for your eyes to accumulate enough reflected photons to see it. If you stop looking at the vase, the vase is still there. If you look at it again, the vase will still be on the table and it will still be green.
I could similarly say that you cannot look at the same vase twice. By the time you measured the vase, the vase doesn't exist any more. Each photon in fact comes from a different vase. So what does "green" actually mean at that point?
This is an irrefutable philosophical position. It all depends on what you define as the same vase, and what measurement means. There are accepted meanings of these words in QM and those meanings coincide with the conventional meanings in the appropriate limit. I'm happy to use different words though, since you're right that those meanings are not exactly the same as the everyday meanings of those words.
> If you doubt this, read the following:
Have you considered the double slit experiment, where the whole point is the difference in length of trajectory?
Note the following facts, consequence of QM:
(1) the fact that if you bloop spin twice that you get the same result twice
(2) the fact that if you bloop a particle with a more or less spherically symmetric wave function (such as arising from a collision) in a bubble chamber, that you get linear tracks
(3) the fact that a single photon can interfere along paths of different lengths
You've painted yourself into a linguistic corner that makes all these phenomena incredibly difficult to understand, and causes you to make statements that contradict experimental evidence under a non-word-play interpretation of the words.
As for point (3), consider that Maxwell's equations are the quantum theory of a single photon. The reason they also work for macroscopic amounts of light, is that the equations are linear and photons only weakly interact with each other. One shouldn't think of a photon as a particle moving in straight lines at the speed of light. A photon is a wave distributed in both space and time. The wavefront tends to move at the speed of light, but you can't think about it as a point particle moving along trajectories. In some cases you can: there are solutions of Maxwells equations in which a wave packet more or less moves in a straight line at the speed of light, and the probability amplitude is more or less zero everywhere else. But the situation being set up in your experiment, or indeed in the double slit experiment, is not of that form.
Therefore, the outcome of your proposed experiment depends on the shape of the wave packet of the photon and the distance difference. If the wave packet is sufficiently localised in space and time, and the distance difference is sufficiently large, then there will be no interference. The wave packet will travel to the half-mirror and split into two, then those two packets will continue traveling such that they never meet again at the same point in space-time (you can see this by visualising the movie of what happens), and there will be no interference. There will be some probability distribution of observing the photon at a given point on the detector at a given point in time, with two peaks in space-time corresponding to the two wave packets.
If however, you decrease the distance difference so that the two wave packets will overlap in space-time again after the bounce, there will be interference.