Commitment schemes with verification in NC0
No. $\;\;\;\;$ There is also no secure commitment scheme where the verification

routine can be done with bounded fan-in in depth $\:o(\hspace{.02 in}\log(\hspace{.02 in}\log(k)))\:$,

since such circuits have input locality $\:(\hspace{.02 in}\log(k))^{o(1)}$,$\:$ which means one can

efficiently brute-force the bits that might affect the verification circuit's output.