  Quantum coherence with respect to orthonormal bases has been studied
extensively in the past few years. Recently, Bischof, et al. [Phys. Rev. Lett.
123, 110402 (2019)] generalized it to the case of general positive
operator-valued measure (POVM) measurements. Such POVM-based coherence,
including the block coherence as a special case, have significant operational
interpretations in quantifying the advantage of quantum states in quantum
information processing. In this work we first establish an alternative
framework for quantifying the block coherence and provide several block
coherence measures. We then present several coherence measures with respect to
POVM measurements, and prove a conjecture on the $l_{1}$-norm related POVM
coherence measure.
