Which is the biggest object which can interfere with itself? I've heard that scientist proof that viruses of  the Tobacco mosaic virus could interfere with themselves. I'm referring to quantum interference-- the same as photons. Unfortunately, I couldn't find any material on this problem on the Internet , so I don't know if it's true. Viruses have a width between 20nm and 1000nm, so they should be applied to mesoscopic physics, where that phenomenon is possible. 
Is there proof that some species of virus can interfere with themselves?
If not, which is the biggest object where we may observe that interference? 
 A: I love versions of this question. So, even though I think you already have a couple of nicely definitive answers above, I can't resist diving in a bit more deeply into the "why" of quantum interference.
For anyone reading this who does not know quite what "quantum interference" means, the simplest possible example is this: It means that an object "in some sense" passes through two or more paths at the same time. The result on the other side is distinctly and visibly different from passing through just one path. If for example you have two paths defined by openings or slits in a large wall, the quantum interference path will if set up the right way be strongest exactly between the two openings or slits. That sounds pretty weird and it is, but it is also exactly how the universe has been shown to work to a truly remarkable degree of precision. When objects exhibit this "I followed more than one path at once" kind of behavior, the shorthand term for it is quantum interference. Quantum interference is very well proven for very small objects like particles of light (photons), electrons, and even atoms, but as you get larger it starts getting very dicey and a lot harder to do. The modern phrase for this unraveling of the quantum interference effect is "decoherence," though I should emphasize that it was not always called that.
As usual, I like to focus the "why" of certain physics effects. In this case, why is it so darned hard to make large objects quantum? So let's look this puppy over a bit, using quantum principles readily available Volume III of the Feynman Lectures on Physics, and intentionally shooting for a big-object case that is really interesting:
What would it take to get a spaceship with a passenger to interfere with itself in the way as an electron in a two-slit experiment?
Why is that one interesting? Well, mostly because it says that based on simple scale extrapolations of experimentally known physics, there is no reason that anyone knows of why you or I couldn't quantum interfere!
That is, there is no known physics reason why a human could not in principle pass through two holes in distant space wall "simultaneously" in exactly the same way as an electron in a lab, provided that two prerequisites that I'll get into below are met. Electrons of course have very few "inner gear" options to work with -- pretty much just spin orientation in fact -- so they are a bit boring in terms of their "memory" of what happens during such a transition. A human in contrast could be thinking or doing many things while at the same time interfering with themselves by apparently passing though more than one region of space at the same time. Talk about split personalities!
So if internal wheels an gears and memory and cognition don't prevent objects from interfering, what does? And why do those things make it harder for larger objects?
The first reason is the best known, and it's simply an issue of frequency. It turns out that the mass-energy of an object has associated with it a frequency that is proportional to the mass. This frequency is very, very high even for tiny objects. For one electron, for example, the corresponding frequency is $1.23559277 × 10^{20}$ hertz. That works out to a rather hefty conversion constant of $1.35639584 × 10^{50} kg^{-1} s^{-1}$ for determining the quantum interference frequency of kilogram-scale objects.
Since the distance you need to travel to be "quantum" and travel through two holes is proportional to that frequency, your first big restriction on making a spaceship quantum is that you'd have to travel a very long distance indeed to interfere with yourself. That number alone so absurdly large for holes that are, say, one spaceship width apart, that you have already eliminated any real possibility of a spaceship going quantum on you. (If someone wants to work out the exact numbers... have fun!)
Because quantum frequencies are so absurdly high, the second restriction often gets overlooked. That's too bad though, since it's the harder one! It's this second restriction that makes things like quantum computing really hard, even for very small objects like electrons.
So what is this second, even more severe restriction? It's this: You must not leave any trace anywhere in the universe of your travels -- not even a single photon's worth. If you leave any such trace, your spaceship ceases to become quantum, and your entire many-lifetimes journey becomes a waste. Now that's severe!
Feynman's Volume III Lectures again provide the needed insight into this issue, specifically the section where he talks about trying to "catch" an electron as it sneaks through one hole or the other. You can do it by bouncing a photon off of it... but as soon as you do, the electron ceases to be quantum, and you instead just have an ordinary electron going through an ordinary hole.
Thus you literally have to keep self-interfering electrons in the dark if you want them to interfere, since even a single photon bouncing off of them makes them ordinary and keeps them from interfering.
But what's really tough on large objects is that even one photon has the same effect on an object that may be kilograms in size, rather than electron sized. That's because it's the information found that kills the interference effect, not the mass of the photon or the number of photons.
And there, finally, is why it's even harder than it looks to make large objects interfere: You must embed them in absolute silence of all forms, keeping them isolated from the universe as a whole for the entire journey, to make them interfere.
An addendum: If you take the above points and apply them to the commonly given version of Schrodinger's cat, what do you get? Is the cat truly quantum? Well... no, definitely not, not even under the most optimistic circumstances. Why? Because information is pouring out of the box at all time, via sound, molecules, and radiation. A true Schrodinger's cat requires absolute and total silence and isolation from the universe. The issues of whether there is an observer or not does not really enter into that setup, even abstractly, since until you get that part of information leakage part of the problem under control you don't have a setup that is quantum enough to make the observer question real.
(Amusingly, Schrodinger constructed the cat example in hopes of disproving by absurdity the need for an observer. I think we can safely say that that one backfired! Similarly, John Bell hoped to disprove entanglement, another point that gets lost in everyone's enthusiasm about how he proved Einstein wrong. He did do that, but the superb irony is that Bell was a huge Einstein fan and had fervently hoped to prove Einstein correct.)
A: Superconductivity is a macroscopic manifestation of quantum mechanics and does display interference effects. In this reference of order of centimeters . As there are kilometer lengths in the superconducting magnets at the LHC at CERN it is possible that the macroscopic size is limited by technology.
SQUIDs use such an interference to measure magnetic fields.
A: http://www.nature.com/ncomms/journal/v2/n4/full/ncomms1263.html
Above is a reference with interference of 6 nm molecules achieved
Here is a proposal to use much larger molecules in space:
http://arxiv.org/abs/1201.4756
Here is a proposal to use larger molecules in an optical interfereometer:
http://arxiv.org/abs/1103.4081
