  The entanglement entropy (EE) of quantum systems is often used as a test of
low-energy descriptions by conformal field theory (CFT). Here we point out that
this is not a reliable indicator, as the EE often shows the same behavior even
when a CFT description is not correct (as long as the system is asymptotically
scale-invariant). We use constraints on the scaling dimension given by the CFT
with SU(2) symmetry to provide alternative tests with two- and four-point
correlation functions, showing examples for quantum spin models in 1+1
dimensions. In the case of a critical amplitude-product state expressed in the
valence-bond basis (where the amplitudes decay as a power law of the bond
length and the wave function is the product of all bond amplitudes), we show
that even though the EE exhibits the expected CFT behavior, there is no CFT
description of this state. We provide numerical tests of the behavior predicted
by CFT for the correlation functions in the critical transverse-field Ising
chain and the $J$-$Q$ spin chain, where the conformal structure is well
understood. That behavior is not reproduced in the amplitude-product state.
