Let $F(G)$ be the algebra of functions on a finite quantum group $G$ (so that $F(G)$ is a finite dimensional $\mathrm{C}^*$-Hopf algebra).

Suppose that $\{p_i:i=0,\dots,d-1\}\subset F(G)$ is a partition of unity, in other words $p^2_i=p_i^*=p_i$, $$p_ip_j=p_jp_i=\delta_{i,j}\,p_i,$$ and

$$\sum_{i=0}^{d-1}p_i=\mathbf{1}_G:=1_{F(G)},$$ the unit of $F(G)$.

Edit: The following condition was added after Konstantinos' answer:

Suppose that a state $\nu\in M_p(G):=\mathcal{S}(F(G))$ has the property that for all projections $q\in F(G)$, there exists $k_q$ such that $\nu^{\star k_q}(q)\neq 0$, where $$\nu\star \nu=(\nu\otimes \nu)\circ \Delta.$$

Suppose furthermore that $\nu\in M_p(G)$ has the property that:

$$\nu(p_i)=\begin{cases}1 & \text{ if }i=1\\ 0 & \text{else}\end{cases},$$

and we also have that, where $\varepsilon\in M_p(G)$ is the counit:

$$\varepsilon(p_i)=\begin{cases}1 & \text{ if }i=0\\ 0 & \text{else}\end{cases}.$$

Furthermore,

$$(\nu\otimes I_{F(G)})\circ \Delta(p_i)=:T_\nu(p_i)=p_{i-1},$$ with $T_\nu(p_0)=p_{d-1}$.

Note that $\Delta$ is a *-homomorphism, and, where $\int_G:=h\in M_p(G)$ is the Haar state of $F(G)$, we can show that:

$$\int_Gp_i=\frac{1}{d}.$$

Is it the case that

$$\Delta(p_i)=\sum_{k=0}^{d-1}p_{i-k}\otimes p_k?$$

If $F(G)$ is commutative, this condition holds.