Title: On Blockwise Symmetric Matchgate Signatures and Higher Domain \#CSP,
Abstract: For any $n\geq 3$ and $ q\geq 3$, we prove that the {\sc Equality} function
$(=_n)$ on $n$ variables over a domain of size $q$ cannot be realized by
matchgates under holographic transformations. This is a consequence of our
theorem on the structure of blockwise symmetric matchgate signatures. %due to
the rank of the matrix form of the blockwise symmetric standard signatures,
%where $(=_n)$ is an equality signature on domain $\{0, 1, \cdots, q-1\}$. This
has the implication that the standard holographic algorithms based on
matchgates, a methodology known to be universal for \#CSP over the Boolean
domain, cannot produce P-time algorithms for planar \#CSP over any higher
domain $q\geq 3$.