tar itself does not write down a checksum for later comparsion. If you gzip the tar archive you can have that functionality.
tar uses compress. If you use the -Z flag while creating the archive tar will use the compress program when reading or writing the archive. From the gzip manpage:
The standard compress format was not designed to allow consistency
checks.
But, you can use the -z parameter. Then tar reads and writes the archive through gzip. And gzip writes a crc checksum. To display that checksum use that command:
$ gzip -lv archive.tar.gz
method crc date time compressed uncompressed ratio uncompressed_name
defla 3f641c33 Sep 25 14:01 24270 122880 80.3% archive.tar
From the gzip manpage:
When using the first two formats (gzip or zip is meant), gunzip checks
a 32 bit CRC.