  Nonstabilizerness, or `magic', is a critical quantum resource that, together
with entanglement, characterizes the non-classical complexity of quantum
states. Here, we address the problem of quantifying the average
nonstabilizerness of random Matrix Product States (RMPS). RMPS represent a
generalization of random product states featuring bounded entanglement that
scales logarithmically with the bond dimension $\chi$. We demonstrate that the
$2$-Stabilizer R\'enyi Entropy converges to that of Haar random states as
$N/\chi^2$, where $N$ is the system size. This indicates that MPS with a modest
bond dimension are as magical as generic states. Subsequently, we introduce the
ensemble of Clifford enhanced Matrix Product States ($\mathcal{C}$MPS), built
by the action of Clifford unitaries on RMPS. Leveraging our previous result, we
show that $\mathcal{C}$MPS can approximate $4$-spherical designs with arbitrary
accuracy. Specifically, for a constant $N$, $\mathcal{C}$MPS become close to
$4$-designs with a scaling as $\chi^{-2}$. Our findings indicate that combining
Clifford unitaries with polynomially complex tensor network states can generate
highly non-trivial quantum states.
