Are there results/techniques pertaining to the analysis of squares of random matrices ?More specifically, let $A$ be an $n\times n$ matrix such that each entry is $1$ or $-1$ independently and with equal probability. Now if we want to analyze for any $u,v \in \{-1,1\}^n$, we can make a case for concentration of the value of $u^TAv$ using chernoff bound arguements. However suppose now we want to analyze the value of $u^TA^2v$. This time due to a lot of dependencies among the variables a chernoff type arguement becomes difficult or at least I cannot see it straightaway. Could someone point me to an analysis for this scenario ?  
Squares of random binary matrices
pr.probability;randomness;matrices;chernoff bound