conservation laws - Where isn't there a quantum number associated with the Lorentz boost?

In Schwartz's QFT, there is a problem (3.2) on p. 42 asking to calculate the conserved current $K_{\mu\nu\rho}$ associated with the Lorentz transformation $x^\mu\rightarrow\Lambda^\mu{}_\nu x^\nu$. It's expression is

$$K^\mu{}_{\nu\rho}=T^\mu{}_{[\rho}x_{\nu]} \tag{1}$$ where, the square brackets imply the antisymmetrization.

One can define the conserved quantities $$Q_j=\int d^3x~K^0{}_{0j},\tag{2}$$ which induce the boosts. Being conserved, one means that $$\frac{dQ_j}{dt}=0.\tag{3}$$

However, Schwartz also stated that $\frac{dQ_j}{dt}=0$ is consistent with $$i\frac{\partial Q_j}{\partial t}=[Q_j,H]~! \tag{4}$$

Now, let us look at the definition of $Q_j$ more carefully. You will find out that $Q_j$ is a function of $t$ only. So $\frac{\partial Q_j}{\partial t}=\frac{dQ_j}{dt}$. But $[Q_j,H]\ne0$! What is going on?