  Many multiprocessor systems have interconnection networks as underlying
topologies and an interconnection network is usually represented by a graph
where nodes represent processors and links represent communication links
between processors. In 2016, Zhang et al. proposed the $g$-extra diagnosability
of $G$, which restrains that every component of $G-S$ has at least $(g +1)$
vertices. As an important variant of the hypercube, the $n$-dimensional crossed
cube $CQ_{n}$ has many good properties. In this paper, we prove that $CQ_{n}$
is tightly $(4n-9)$ super 3-extra connected for $n\geq 7$ and the 3-extra
diagnosability of $CQ_{n}$ is $4n-6$ under the PMC model $(n\geq5)$ and MM$^*$
model $(n\geq7)$.
