Orangealpine's last post is exactly correct.
And Entropy, if you wish to speak with authority on a subject, then make sure you master it before you do, especially when calling it "hogwash".
Absolute thermodynamic temperature is defined as follows through the Boltzmann distribution:
N/N° = e^(-deltaE/kT)
This signifies that the number of molecules at a higher energy level +deltaE , N, with respect to those at a lower energy level, N°, uniquely defines a property "T", temperature, with the aid of a constant k, Boltzmann's constant, of which the value is well-known. Through statistical thermodynamics also entropy follows from this proportion.
At low temperature, the proportion of high-energy particles will be very low. As temperatures get higher and higher, the ratio will go up, until, at infinite temperature, the proportion will be exactly 1, and there will be equal populations at all energy levels. Entropy, at the same time, will rise.
Now imagine what happens with the mathematics if you succeed in creating a system where the higher energy level has a higher population than the lower energy level (that's your "infinity plus one degree").
Exactly - the expression at the right hand side will go positive, implying that the temperature has now gone formally negative. Entropy, in this "virtual" temperature range, behaves exactly the same as it does in the real temperature range - if the temperature rises to approach zero (which is not the same zero as in the positive temperature range, as here the population between low and high energy levels would be completely inverted) the entropy lowers.
You can think through the consequences yourself - how will this system behave when left to its own devices ? The high energy particles will lose energy, and the system will formally "cool down" quickly and its entropy will rise ... from negative temperatures, to very negative temperatures, and finally it will reach an asymptote at negative infinity, at which point it will switch to positive infinity and cool down further from there, with the entropy now lowering.
So from the mathematics, negative temperatures have nothing at all to do with "zero movement" and "collapsed electrons" - they are states with an internal energy higher than infinity.
Impossible, you say ? Well, apparently these dudes have succeeded in creating some. Which is why they are published in Science in the first place.
The impossible is only defined through the limits of one's imagination.