Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of a regular local ring by a regular sequence). The information that I allow myself is the knowledge of the Hilbert series, that is, of the function $$n \mapsto \mathrm{length}_R (R/\mathfrak{m}^n).$$ Is such a thing possible? More precisely, is there a criterion for $R$ to be a complete intersection in terms of the Hilbert function? More loosely, is there a sufficient (but perhaps not necessary) condition on the Hilbert function that would ensure that $R$ is a complete intersection (I am not interested in criteria that also force regularity of $R$)?

If a criterion for complete intersection is not possible, I would be willing to relax the assumptions: I would also be interested in criteria for $R$ to be Gorenstein or merely Cohen-Macaulay. Although I have found some literature on similar questions in the case when $R$ is assumed to be graded at the outset, I am not really interested in the graded case.