Can we entangle particles on "large" distances? So far I've only encountered statements, that two particles that we want to entangle have to be really close to each other. But is the entanglement of these quantum entities possible on "larger" distnces?
Two particles can entangle if we have a Hamitlonian of specific nature. So, at first sight I see no obstacles to design such an experiment (at least in theory), which implments the desired interactions between particles that are separated by some distance.
What do you think about it?
 A: Particles that are already entangled can be brought to large distances from one another and remain entangled. This means that measurements on each of the two particles will show statistical correlations (IE my result here has a statistical relation to what you chose to measure over their) that one can prove would not be possible with a so-called "local hidden variable model". These statistical trends are not enough to send signals faster than light.
As for getting the particles entangled in the first place, I am not aware of any forces that can occur at a distance without an exchange particle (indeed if their were you COULD send signals faster than light). So their should be no physical Hamiltonian that is nonlocal (acts on things at two places at once). 
Their might be some Hamiltonains that people use in models that appear non-local (the Ising model Hamiltonian for instance: https://en.wikipedia.org/wiki/Ising_model, has an interaction between adjacent spins), but these will always be approximations. For example the true Hamiltonian for the spin-spin force should include the exchange photon that propagates between the spins.

Add on later:
It is actually possible to entangle two particles that have never interacted with one another (even indirectly). If particle A is entangled with particle B (A-B) and particles C and D are entangled (C-D), it is possible to perform a procedure called "entanglement swapping" where you get particles B and D, perform some process, and arrive in the state where A and C are entangled (A-C).
This can be done even if A and C were never anywhere near eachother, and even if they have never interacted with one another (even indirectly).
This was the approach used in: https://www.nature.com/articles/nature15759
A: The entanglement doesn't have to be initiated when the particles are already far apart. You can entangle two particles and then separate them. Once the particles are entangled, the entanglement can be amplified and spread to other objects. This has been done with macroscopic objects at macroscopic distances (15 cm):
K.C. Lee et al.,  "Generation of room-temperature entanglement in diamond with broadband pulses," Oxford (PhD thesis), 2012, 
https://www2.physics.ox.ac.uk/research/ultrafast-quantum-optics-and-optical-metrology/theses-0
This paper has a nice discussion of Lee's work and a nice simple explanation of the physics: https://arxiv.org/abs/1601.07927
For individual particles rather than macroscopic objects, I believe entanglement has been demonstrated at much larger distances, like kilometers, but I don't have references handy. Quantum mechanics doesn't have any built-in time or distance scales, since Planck's constant doesn't have the right units to determine such a scale, so there is no limit to how large the distance can be.
Entanglement can also be created over spacelike intervals in spacetime:
Guerreiro et al., 2012, http://arxiv.org/abs/1204.1712
A: If you look at the fundamental forces table here you will see that is only the electromagnetic interaction that has the range  to  large distances:

The other is the gravitational, which has a very much smaller coupling than the electromagnetic.
The coulomb potential which enters into the Shrodinger equation for two elementary charged particles will be microscopically small at large distances. This means that the probability of scattering at such distances will be effectively zero .
In order to have entanglement,one wavefunction should describe the system. To be able to induce entanglement at large distances it should be possible to have measurable interactions, in this case scattering, but the probability is very very low.
Even though in principle the underlying framework is quantum mechanical,  it will not be possible to measure any correlations at these large distances.
