Assigning a chaos index for dynamics of generic quantum field theories is a<br />
challenging problem, because the notion of Lyapunov exponent, which is useful<br />
for singling out chaotic behaviors, works only in classical systems. We address<br />
the issue by using the AdS/CFT correspondence, as the large $N_c$ limit<br />
provides a classicalization (other than the standard $\hbar \to 0$) while<br />
keeping nontrivial quantum condensation. We demonstrate the chaos in the<br />
dynamics of quantum gauge theories: Time evolution of homogeneous quark<br />
condensates $\langle \bar{q}q\rangle$ and $\langle \bar{q} \gamma_5 q\rangle$<br />
in an ${\cal N}=2$ supersymmetric QCD with the $SU(N_c)$ gauge group at large<br />
$N_c$ and at large &#039;t Hooft coupling $\lambda \equiv N_c g_{\rm YM}^2$ exhibits<br />
a positive Lyapunov exponent. The chaos dominates the phase space for energy<br />
density $E \gtrsim (6\times 10^2)\times m_q^4(N_c/\lambda^2) $ where $m_q$ is<br />
the quark mass. We evaluate the largest Lyapunov exponent as a function of<br />
$(N_c,\lambda,E)$ and find that the ${\cal N}=2$ supersymmetric QCD is more<br />
chaotic for smaller $N_c$.