That's not true. Even if there was such a thing as a "position" of a particle (hint: look up "wavefunction"), it does not "jump". It evolves following the time-dependent Schrödinger equation. You/Alpha may be confused because a bound particle (e.g. an electron in an atom) seems to "jump" between stationary solutions of the equation ("orbitals"). But there is a smooth transition between these states. On the other hand, a free particle will travel in a continuous fashion.
By the way, if you can demonstrate that space is quantized, you may get a Nobel. Before that, it's just statements without arguments.
ETA: your link was a nice read, but it does not support your/Alpha's argument.
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