  We study quasimorphisms and bounded cohomology of a variety of braided
versions of Thompson groups. Our first main result is that the Brin--Dehornoy
braided Thompson group $bV$ has an infinite-dimensional space of quasimorphisms
and thus infinite-dimensional second bounded cohomology. This implies that
despite being perfect, $bV$ is not uniformly perfect, in contrast to Thompson's
group $V$. We also prove that relatives of $bV$ like the ribbon braided
Thompson group $rV$ and the pure braided Thompson group $bF$ similarly have an
infinite-dimensional space of quasimorphisms. Our second main result is that,
in stark contrast, the close relative of $bV$ denoted $\hat{bV}$, which was
introduced concurrently by Brin, has trivial second bounded cohomology. This
makes $\hat{bV}$ the first example of a left-orderable group of type
$\operatorname{F}_\infty$ that is not locally indicable and has trivial second
bounded cohomology. This also makes $\hat{bV}$ an interesting example of a
subgroup of the mapping class group of the plane minus a Cantor set that is
non-amenable but has trivial second bounded cohomology, behaviour that cannot
happen for finite-type mapping class groups.
