  Arithmetic complexity is considered simpler to understand than Boolean
complexity, namely computing Boolean functions via logical gates. And indeed,
we seem to have significantly more lower bound techniques and results in
arithmetic complexity than in Boolean complexity. Despite many successes and
rapid progress, however, challenges like proving super-polynomial lower bounds
on circuit or formula size for explicit polynomials, or super-linear lower
bounds on explicit 3-dimensional tensors, remain elusive.
  At the same time, we have plenty more "barrier results" for failing to prove
basic lower bounds in Boolean complexity than in arithmetic complexity. Finding
barriers to arithmetic lower bound techniques seem harder, and despite some
attempts we have no excuses of similar quality for these failures in arithmetic
complexity. This paper aims to add to this study.
  We address rank methods, which were long recognized as encompassing and
abstracting almost all known arithmetic lower bounds to-date, including the
most recent impressive successes. Rank methods (or flattenings) are also in
wide use in algebraic geometry for proving tensor rank and symmetric tensor
rank lower bounds. Our main results are barriers to these methods. In
particular,
  1. Rank methods cannot prove better than $\Omega_d (n^{\lfloor d/2 \rfloor})$
lower bound on the tensor rank of any $d$-dimensional tensor of side $n$. (In
particular, they cannot prove super-linear, indeed even $>8n$ tensor rank lower
bounds for any 3-dimensional tensors.)
  2. Rank methods cannot prove $\Omega_d (n^{\lfloor d/2 \rfloor})$ on the
Waring rank of any $n$-variate polynomial of degree $d$. (In particular, they
cannot prove such lower bounds on stronger models, including depth-3 circuits.)
