The explanation in the video is mostly garbage. (The "what the bleep" title card is a bit of a giveaway.)
The main problem in the video is a statement like "a photon must decide whether it is a particle or a wave." That is a fundamental misunderstanding of quantum mechanics. Photons are quantum mechanical objects whose behavior is, in some circumstances, subject to non-local correlations in the same way as classical waves in a classical fluid can interfere with each other. In other circumstances, the behavior of a quantum-mechanical photon localized and quantized (a word which I use to mean "it comes in a lump") in the same way as a classical particle like a shotgun pellet or a brick.
The counterintuitive feature of quantum mechanics is that the non-local, wave-like features of the photon field include correlations among the particle-like behaviors. For example, the rate at which photons interact (one at a time, like "particles") with position-sensitive photon-absorbing detectors depends on the intensity of the photon field, which you can vary by changing the amount of energy stored in a laser cavity. Suppose you have some interferometer which is a foot across, so that a particle traveling at lightspeed would take a nanosecond to cross it, and you turn down your photon intensity so that you have one photon per second interacting with your detector. If photons were countable, like shotgun pellets, that would mean that usually there are zero photons in your apparatus; there would be one photon in your apparatus very briefly once per second, on average; the probability that there would be two photons in your detector simultaneously would be roughly a billion to one. Each single photon shows up in a position-sensitive detector at exactly one location. And yet, adding up the locations of those "single" photons reveals an intensity distribution that's exactly the same as you would predict using wave interference (e.g.). Each "single" photon apparently interacts with the entire apparatus. People who don't like non-local interactions are upset by this.
In the delayed-choice eraser experiment we take this a step further by having three opporunities for the photon field to behave non-locally:
- A photon from a laser passes through a double slit.
- After the double slit, the photon is converted into an entangled pair of photons. One, the "signal," goes to a position-sensitive detector.
- The "idler" entangled photon goes to an interferometer, where it take take one of four paths. Two of the paths definitely correspond to one or the other of the slits in phase 1; the other two are mixed and can come from either of the initial slits.
After running the experiment for a while, you have a large number of position measurements from the signal photon which don't show any interference pattern. Each signal photon is associated with an idler photon which took one of the four paths through the second interferometer. If you throw away all of the signal photons except where the idler photon took a path through the interferometer that corresponds to a particular slit, you still see no interference pattern in the signals' positions. If you look at the signal photons from where the idler photon took either of the routes through the interferometer --- that is, the other half of your data --- you also see no interference pattern in the signals' positions. But if you choose just the data where the idler photon took one of the ambiguous paths through the interferometer --- that is, a quarter of your data --- an interference pattern appears in the position data for those signal photons. (If you choose the other ambiguous path, you get a complementary interference pattern; everyone is accounted for.)
What upsets people who don't like non-local interactions is that you can adjust the order of the signal and idler photons, so that the signal photon is detected (and destroyed) before the idler photon has a chance to interact with the final interferometer, and you still see the same interference effect.
There's a lot of controversy about exactly how to interpret these results. The explanation in your linked video, in which the idler photon's choice of path in the interferometer goes back in time and changes whether the signal photon "was a particle or a wave", is garbage for several reasons, some of which I addressed above --- but it's so provocative that people respond to it, and repeat it, without examining the more complicated reality. I've tried to write the answer using the language the I prefer, which is that the photon field is fundamentally non-local, including correlations among different detectors.
theoretically you can have a device with photon container in which you store the photon for a significant amount of time. Then in future you can measure a photon stored in device or destroy device without measuring. If you measure it you destroy interference pattern in past in the moment photon's twin was released. It means you can send 1 or 0 in past from future which in turns means you can send any kind of data
But that's not correct. The "choice" which "goes back in time" (note the scare quotes) is a choice that the idler photon makes when it interacts with the final interferometer. You can't make that choice on its behalf --- the choice is made by the way that the entire photon state, including both entangled photons, interacts with the entire apparatus. The high-quality wikipedia particle on the delayed-choice quantum eraser includes the sentence
[A] theorem proved by Phillippe Eberhard shows that if the accepted equations of relativistic quantum field theory are correct, it should never be possible to experimentally violate causality using quantum effects.
which cites this and this reference.