It is logical to think about a quantized time? I have always wondered if it makes sense to think about time like something discrete. In this way time would work similarly to an hourglass, where every sand grain would be an amount of time; the bigger is the grain and the bigger is the amount of time carried by a single grain; like in photons the higher is the frequency and the higher is the energy carried by a single photon.
And when we are in an high gravitational field maybe the "sand grains" are smaller  proportionally to the intensity of the field.
Assume a second is carried by two "small grains" or one "big grain".
I am near an high gravitational field. When I felt a second, two small grains fall down. 
The other person in the empty space felt a second when one big grain has fallen down. 
But the grains fall down together. So the first of my small grain fall down with the big grain of the other person. The result is that when for the other person in the empty space pass a second, for me pass only 1/2 second (carried by one small grain) and so on. 
In the same way we can think that when increasing our velocity the grains become smaller until they disappear (radius=0) when we reach the light speed. So this velocity can't be reached because the grains need to fall down together in the whole universe and it is impossible if they disappear. 
Is all that logical? I'm not a physicist so maybe I'm saying something completely incorrect. 
 A: I do want to try an answer regarding the title of your question.

It is logical to think about a quantized time?

Temporarily leaving aside the rest of your post, it is logical and perfectly  "legitimate" to consider time, if you prefer actually "spacetime" in more general terms, as potentially being of a discrete composition.
Tests have been carried out on cosmological data, for example radio and light waves, to check if, over very long distances, we can detect any changes to what we would expect if time was discrete, and so far we have failed to detect any definitive evidence for a  discrete view of time.
For physics, the definition of time is that which is measured by a clock. Any modification or variation of this simple measure invariably  leads to philosophical discussions on the nature of time, so physicists take the clock definition as a pragmatic and more importantly,   measureable view of time.
If time / spacetime was discrete, it might help solve many current physical problems, whilst at the same time creating a host of new ones, depending on which area of physics you are working in.

I have always wondered if it makes sense to think about time like something discrete. In this way time would work similarly to an hourglass, where every sand grain would be an amount of time; the bigger is the grain and the bigger is the amount of time carried by a single grain; like in photons the higher is the frequency and the higher is the energy carried by a single photon. And when we are in an high gravitational field maybe the "sand grains" are smaller proportionally to the intensity of the field.

Your question then asks about how logical your personal idea is. I cannot answer that for various reasons.


*

*Personal theories are off limits here, because well,  they are personal and usually not testable, yours is not testable and you do not suggest any tests.

*I go back to physics definition of time as that measured by a clock, and since this works very well for physics,  we will stick with it, thanks all the same:)  no matter what personal theory comes along, especially untestable ones.
To sum up, the answer to the title of your question is yes.
The answer to the question in the text of your post is: give us a method of testing your theory, or a math based version of it, and physics might feel it is worth pursuing. 
But in the form your  question is in, despite the effort you have put into it, unless you convert it from, no offence intended, musings, into one or both of  the forms mentioned in the paragraph above, the physics community will not be able to make any judgement on your ideas. 
