For a probability distribution $X$, supported on integers, define the shift-invariance of $X$, denoted by $shift(X)$ = total variation distance between the random variable $X$ and $X+1$.
Let $p:\mathbb{R}^n \rightarrow \mathbb{R}$ be a homogenous multilinear polynomial of total degree $2$ and the coefficients of $p$ come from the set $\{0,1\}$. Consider the random variable $Z = p(x_1, \ldots, x_n)$ where each $x_i$ is an independent unbiased $\{0,1\}$ random variable. Do we know of some property of $p$ which will ensure that $shift(Z)$ is small? For example, if $p$ were instead linear, then if the number of terms in $p$ is $k$, then $shift(Z) \le O(1) \cdot \sqrt{1/k}$ ?
