Reason by analogy to Aharonov-Bohm experiment. [Aharonov-1959], [Aharonov-2005].
Look at paths around a solenoid, travelling to either side of it, but neither experiencing any magnetic field.
Travel through a vector potential however, so that the term will create a phase shift.
Working to first order here (Aharonov Bohm paper more sophisticated, sums over loops). Use constant potential kernel from semi-classical approximation above.
To get phase shift of wave function to lowest order, integrate pA term along the path.
Get a phase correction corresponding to line integral of vector potential:
This is the heart of semi-classical approximation.
The integral over the vector potential is a function of the magnetic field within the solenoid, even though there is no magnetic field anywhere on the path.
Therefore changing the value of the magnetic field, induces changes in the relative phase shift of the two paths, and therefore changes in the phase shift at the detector.
Extraordinary.
Further, the effect is essential if quantum mechanics is to be self-consistent; not an outlier but at the heart of quantum mechanics's mysteries.
This discussed in Aharonov-Bohm.
As noted above, can often get a temporal quantization experiment from a given standard quantum theory one by interchanging a time and space axis.
This also flips magnetic and electric fields.
So we expect a temporal quantization equivalent to the Aharonov-Bohm experiment experiment might use electric rather than magnetic fields.
[ flip ]
Setup consists of a beam we can split in two, a capacitor we can turn on and off, and a pair of delay loops, one in front of the capacitor, one behind it.
The delay loop has to "park" an electron or other charged particle for a specified (laboratory) time without loss of coherence.
Now, while the capacitor is off we split the beam in two and send half of it through a hole in the capacitor, while reserving the rest. The near and far sides of the beam are then fed into delay loops. Then, the capacitor is turned on. There is no field outside the capacitor (it is an ideal capacitor) so neither particle sees a field. However, there is a potential difference between the two sides, so one particle sees a potential of V relative to the other.
[ cite semi-classical approximation above, for constant potential ]
Therefore one particle experiences a phase change given by the integral:
Now the capacitor is turned off and the nearside beam sent through the hole in the capacitor to be recombined with the farside of the beam.
They will interference destructively or constructively depending on the relative phase change. The relative phase change can be tuned by changing the voltage on the capacitor and the amount of time the capacitor is turned on.
If the potential is V on the near side and zero on the far side, then the phase shift is:
On right is zero.
As it happens, we see essentially the same effect in standard quantum theory. Non-relativistic Lagrangian gives
Effect present in standard quantum theory and temporal quantization both; quantitatively but not qualitatively different.
Since Aharonov and Rohrlich show the effect needed for self-consistency of quantum mechanics in case of vector potential, expect needed for consistency of quantum mechanics treatment in electric potential as well.
So, Aharonov-Bohm experiment in time does not discriminate decisively between temporal quantization and standard quantum theory. But does suggest that temporal quantization offers a useful way to develop new experiments in time.