  We show the properties and characterization of coherence witnesses. We show
methods for constructing coherence witnesses for an arbitrary coherent state.
We investigate the problem of finding common coherence witnesses for certain
class of states. We show that finitely many different witnesses $W_1, W_2,
\cdots, W_n$ can detect some common coherent states if and only if
$\sum_{i=1}^nt_iW_i$ is still a witnesses for any nonnegative numbers
$t_i(i=1,2,\cdots,n)$. We show coherent states play the role of high-level
witnesses. Thus, the common state problem is changed into the question of when
different high-level witnesses (coherent states) can detect the same coherence
witnesses. Moreover, we show a coherent state and its robust state have no
common coherence witness and give a general way to construct optimal coherence
witnesses for any comparable states.
