It is an old theorem of Heaton's (based on work of Eilenberg and MacLane), that a polynomial 3cocycle $f(x,y,z)$ which is "symmetric," in the sense that $f(x,y,z)f(x,z,y)+f(z,x,y)=0$, is always a coboundary. Does anyone know of a similar theorem for polynomial 4cocycles? The proof of the theorem for 3cocycles doesn't seem to be written down in one clear piece anywhere, so I've had a tough time just generalizing it, thus a good reference for a modern treatment of even that case would be much appreciated as well.

Heaton showed that if the field $K$ has characteristic zero, then all polynomial cocycles are coboundaries. In characteristic $p$ this is not always the case, but the following result holds (which you may know already). THEOREM: For $n> 2$ and $K$ a field of characteristic $p$, every polynomial $n$cocycle over $K$ of degree less than $p$ in the first indeterminate and less than $p1$ in the second is a coboundary. It is not possible to relax the requirements given in the above theorem on the degrees; there are counterexamples, see R. Heaton: A Note on Polynomial Cocycles, Math. Zeitschr. 76, 2357239 (1961). For a modern application, see for example chapter $1$ on formal groups (and symmetric polynomial cocycles) in the lecture notes on Shimura Varieties (class of Robert Kottwitz), see http://www.math.ou.edu/~sspallone/ under "Other Writings". 

