  In the standard Bell scenario, when making a local projective measurement on
each system component, the amount of randomness generated is restricted.
However, this limitation can be surpassed through the implementation of
sequential measurements. Nonetheless, a rigorous definition of random numbers
in the context of sequential measurements is yet to be established, except for
the lower quantification in device-independent scenarios. In this paper, we
define quantum intrinsic randomness in sequential measurements and quantify the
randomness in the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality
sequential scenario. Initially, we investigate the quantum intrinsic randomness
of the mixed states under sequential projective measurements and the intrinsic
randomness of the sequential positive-operator-valued measure (POVM) under pure
states. Naturally, we rigorously define quantum intrinsic randomness under
sequential POVM for arbitrary quantum states. Furthermore, we apply our method
to one-Alice and two-Bobs sequential measurement scenarios, and quantify the
quantum intrinsic randomness of the maximally entangled state and maximally
violated state by giving an extremal decomposition. Finally, using the
sequential Navascues-Pironio-Acin (NPA) hierarchy in the device-independent
scenario, we derive lower bounds on the quantum intrinsic randomness of the
maximally entangled state and maximally violated state.
