What are "interferences of higher order" in the context of Born rule and triple-slit diffraction? This question relates to the paper commented in
this 2010 article. 
The paper itself is Ruling Out Multi-Order Interference in Quantum Mechanics; it is the discussion of a triple-slit interference experiment testing the validity of Born rule. 
The abstract asserts that 

Born’s rule predicts that quantum interference, as shown by a double
  slit  diffraction experiment, occurs from pairs of paths.

I do not understand this statement. 
The theory behind it originates from Quantum Mechanics as Quantum Measure Theory (Sorkin, 1994) where is is said 

Why are probabilities squares of amplitudes; why are they expressed
  most naturally in terms of pairs of paths rather than individual
  paths?

Why indeed? If we refer to the path integral description all possible paths are taken into account, including the very weird ones, and there is an infinity of them. What is the point in decomposing the interference pattern of a three-slit diffraction in terms of pairs of paths ?
When the first article says

Born's rule is one of the key laws in quantum mechanics and it
  proposes that interference occurs in pairs of possibilities.
  Interferences of higher order are ruled out.

what does it means exactly? Is there something here that gives any real insight about Born rule, or is this way of picturing interferences just the effect of a specific mathematical treatment, while being actually equivalent to the path integral formalism?
More generally, what is the significance of this experiment?
 A: I'm just guessing here, but one way to interpret it goes like this.
If you have the amplitude $\Phi$ for an outcome that can be achieved in multiple indistinguishable ways each described by an amplitude $\phi_i$ with $i \in 1, 2, \dots , n$ we write
$$\Phi = \sum_{i=1}^n \phi_i \,.$$
That makes the probability for the outcome
\begin{align*}
 P &= \Phi^* \Phi\\
&= \left( \sum_{i=1}^n \phi_i \right)^* \left( \sum_{i=1}^n \phi_i \right) \\
&= \left( \sum_{i=1}^n \phi_i^* \phi_i \right) + \left[ \sum_{i=1}^{n-1} \sum_{j=i+1}^n \left( \phi_i^* \phi_j + \phi_j^* \phi_i\right)\right] \,.
\end{align*}
The single sum is over the squares of the individual amplitudes each of which is non-negative. All the interference comes from the cross-terms each of which involves a pairing of terms from exactly two possible ways for the outcome to be achieved.
So, no single term involving more than two ways of achieving the outcome appears in the probability of that outcome, and I suppose that this is what is referred to. But I don't understand what significance the authors ascribe to the fact.
A: Basically, interference of higher order means an interference term that involves three or more slits.
In the double slit experiment, according to Born's rule,
$$\lvert A+B\rvert^2 = \lvert A\rvert^2 +\lvert B\rvert^2 + A^*B + B^*A$$
$$\lvert A+B\rvert^2 = \lvert A\rvert^2 +\lvert B\rvert^2 + I_{AB}$$
Notice that the interference term $I_{AB}$ involves the two slits $A$ and $B$.
For three slits, according to Born's rule,
$$\lvert A+B+C\rvert^2 = \lvert A\rvert^2 +\lvert B\rvert^2 + \lvert C\rvert^2+ I_{AB} + I_{AC} + I_{BC}$$
Notice that there is no interference term that involves A, B and C all at once, i.e. there is no $I_{ABC}$. 
In this 2010 article, they used the fact that the probability for three slits $P_{ABC}$, could also be expressed as:
$$\lvert A+B+C\rvert^2 = P_{ABC} = P_{AB} + P_{AC} + P_{BC} - (P_{A} + P_{B} +P_{c})$$
$$I_{ABC} = P_{ABC} - (P_{AB} + P_{AC} + P_{BC}) + P_{A} + P_{B} +P_{c}$$
They measured all seven quantities in the right side and concluded that there was no Born rule violation. In the paper, they provided an upper bound for $I_{ABC}$ which is $10^{-2}$ of the expected pairwise interference. 
The significance of the experiment is that it provided an upper bound for the possible deviations from Born rule. In other words they concluded that Born rule is true within their bound of experimental errors.
