\section{Appendix}

\emph{Appendix A: Measurement settings of Alice and Bob in setup 1.}---Let the following six unit vectors $\vec b_x$ ($x=1,\ldots,6$) correspond to Bob's measurement settings with outcome $b=0$ and their antipodal points $-\vec b_x$ to the outcome $b=1$:
\begin{align}
\vec b_1 &= [0, 1, \phi]/n, \quad\vec b_2 = [0, 1, -\phi]/n,\nonumber\\
\vec b_3 &= [1, \phi, 0]/n, \quad\vec b_4 = [1, -\phi, 0]/n,\nonumber\\
\vec b_5 &= [\phi, 0, 1]/n, \quad\vec b_6 = [\phi, 0, -1]/n,
\label{vecbx}
\end{align}
where $\phi= (1+\sqrt{5})/2$ is the golden ratio and $n=\sqrt{1+\phi^2}$ is a normalization constant. In particular, the $\pm \vec b_x$ vectors above form the vertices of the regular icosahedron. See also figure~\ref{fig_blochs} for the arrangement of the $\vec b_x$ vectors on the Bloch sphere (the respective endpoints are marked by small blue circles). 

On the other hand, Alice's Bloch vectors $\vec a_x$ corresponding to the outcome $a=0$ look as follows 
\begin{align}
\vec a_1 &= [0.1, 0.7664,   -0.6346],\nonumber\\
\vec a_2 &= [-0.1,  0.7664,    0.6346],\nonumber\\
\vec a_3 &= [0.4959,  0.8683,    0.0099],\nonumber\\ 
\vec a_4 &= [0.4959, -0.8683,    0.0099],\nonumber\\
\vec a_5 &= [0.9563,  0,        -0.2925],\nonumber\\
\vec a_6 &= [0,       0,         0].
\label{vecax}
\end{align}
In Fig.~\ref{fig_blochs}, the endpoints of the six $\vec a_x$ vectors are denoted by small red circles. 

\emph{Appendix B: Numerically obtained cyclic EPR steerable states in setup~2.}---We define the state $\ket{\psi_1}=\sum_{i,j,k=0}^{1}c_{ijk}\ket{ijk}$ according to Eq.~(\ref{psi1cijk}), which is used to define the state $\rho(p)$ in Eq.~(\ref{rho3q}). Here we set $p=1$, so we simply have
\begin{equation}
\rho_{ABC}\coloneqq \rho(1)=\frac{\sum_{i=1}^3 \ket{\psi_i}\bra{\psi_i}}{3},
\label{rho3peq1}
\end{equation}
where $\ket{\psi_2}$ and $\ket{\psi_3}$ are derived from  one- and two-qubit translations of $\ket{\psi_1}$ according to Eq.~(\ref{rightS}). Note that the reduced two-qubit states of $\rho_{ABC}$ above have the properties $\rho_{AB}=\rho_{BC}=\rho_{CA}$. Therefore, if we find that $\rho_{AB}$ is one-way steerable, then it also holds true for the other two marginals $\rho_{BC}$ and $\rho_{CA}$, which implies the existence of a tripartite cyclic EPR steerable state. We use the metric $\Delta(\rho_{AB})=R^{\rm in }(\rho_{BA})-R^{\rm out}(\rho_{AB})$ to the quality of the cyclic steerability property of the state~(\ref{rho3peq1}), where the values of the feasible critical radii are restricted to $R^{\rm in }(\rho_{BA})\ge 1$ and $R^{\rm out}(\rho_{AB})<1$. Then according to the conditions~(\ref{R1R2}), $\Delta(\rho_{AB})>0$ implies that $\rho_{AB}$ is one-way steerable for projective measurements and this in turn implies that $\rho_{ABC}$ is a cyclically steerable state. In what follows, we present three such states, denoted $\rho_{ABC}^{(i)}$, where each state is defined by $\ket{\psi_1^{(i)}}$ for $i=1,2,3$ in Eq.~(\ref{rho3peq1}).

Our first state $\rho_{ABC}^{(1)}$ corresponds to the state~(\ref{optpsi1simp}) in the main text. The coefficients $c_{ijk}$ of this state are also shown in the second column of Table~\ref{tab1}. Here we set $c_{111}=0$ and the other seven coefficients are real-valued. The corresponding reduced two-qubit state $\rho_{AB}^{(1)}$ (and its swapped state $\rho_{BA}^{(1)}$) has the following critical radius parameters 
\begin{align}
R^{\rm out}\left(\rho_{AB}^{(1)}\right)&=0.99822006,\nonumber\\
R^{\rm in}\left(\rho_{BA}^{(1)}\right)&=1.0000028,
\end{align}
resulting in $\Delta(\rho_{AB}^{(1)})=0.00178274$. By direct calculation, the negativity of this state is $E_N=0.0630$.

Our next state $\rho_{ABC}^{(2)}$ corresponds to $\ket{\psi_1^{(2)}}$ in Table~\ref{tab1} (third column). This is the best real-valued solution we have found, which has the following critical radius parameters: 
\begin{align}
R^{\rm out}\left(\rho_{AB}^{(2)}\right)&=0.99999963,\nonumber\\
R^{\rm in}\left(\rho_{BA}^{(2)}\right)&=1.00237752,
\end{align}
with $\Delta(\rho_{AB}^{(2)})=0.00237789$. On the other hand, the negativity of this state is $E_N=0.0679$.

Finally, the best complex-valued solution corresponding to the state  $\ket{\psi_1^{(3)}}$ is shown in the fourth column of Table~\ref{tab1}. This solution has the following critical radius parameters 
\begin{align}
R^{\rm out}\left(\rho_{AB}^{(3)}\right)&=0.99999898,\nonumber\\
R^{\rm in}\left(\rho_{BA}^{(3)}\right)&=1.0028066,
\end{align}
with $\Delta(\rho_{AB}^{(3)})=0.00280762$. The negativity of this two-qubit state is $E_N=0.0600$.

\begin{table}[t!]
	\begin{center}
		\begin{tabular}{|l|r|r|r|}
			\hline
			$c_{ijk}$ & $\ket{\psi_1^{(1)}}$ & $\ket{\psi_1^{(2)}}$ & $\ket{\psi_1^{(3)}}$\\
			\hline
			\hline
			$c_{000}$ &   $1$    &    $1$&      $1$\\
			\hline
			$c_{001}$ &   $-0.321193$    &    $-0.259910$&      $-0.252592-0.065698i$\\
			\hline
			$c_{010}$ &   $-0.477021$    &    $-0.591007$ &     $0.002913-0.000635i$\\
			\hline
			$c_{011}$ &   $0.045221$  &     $0.028007$ &     $0.025469+0.025479i$\\
			\hline
			$c_{100}$ &   $-0.718592$  &     $-0.798924$ &     $-0.120348-0.110323i$\\
			\hline
			$c_{101}$ &   $0.213715$  &     $0.206125$ &     $-0.103340-0.161335i$\\
			\hline
			$c_{110}$ &   $-0.0482$  &     $-0.079214$ &     $-0.044067-0.089806i$\\
			\hline
			$c_{111}$ &  $0$  &     $-0.000311$ &     $0.055929-0.044192i$\\
			\hline
		\end{tabular}
	\end{center}
	\caption{
		Table for the coefficients $c_{ijk}$ of $\ket{\psi_1^{(i)}}$ in Eq.~(\ref{psi1cijk}) for $i=1,2,3$.}
	\label{tab1}
\end{table}

\emph{Appendix~C: Entanglement properties of the numerically obtained cyclic EPR steerable states in scenario 2.}---Here we analyze the tripartite entanglement properties of the cyclic EPR-steerable states $\rho_{ABC}^{(i)}$, $i=1,2,3$ given in Appendix~B. In particular, we use the criteria of genuine tripartite entanglement (GTE) developed by G\"uhne and Seevinck~\cite{GS10} and the numerical method developed in Ref.~\cite{JMG11}. On the one hand, we have already given the negativity $E_N$ of the two-qubit reduced states of $\rho_{ABC}^{(i)}$ in Appendix~B. Since all $E_N(\rho_{AB}^{(i)})$ are strictly positive, it entails that all two-qubit marginals of $\rho_{ABC}^{(i)}$ are entangled. However, to the best of our knowledge, it is an open problem whether a given three-qubit state with all two-qubit marginals entangled implies that the three-qubit state is genuinely tripartite entangled (GTE). Indeed, multipartite entanglement properties can be very complex even for simple multipartite states, such as three-qubit states~\cite{horo_review,gt_review,szilard}.

Here we show that the complex-valued three-qubit state $\rho_{ABC}^{(3)}$ given in Appendix~B is genuinely tripartite entangled. To this end, we first define biseparable states. A three-party state $\rho_{ABC}$ can be written in biseparable form if and only if:
\begin{equation}
\rho_{ABC}=p\rho_{A|BC}+q\rho_{B|AC}+(1-p-q)\rho_{C|AB},
\label{bisep}
\end{equation}
where $p,q,(1-p-q)$ are non-negative numbers and $\rho_{A|BC}$ represents any tripartite biseparable state with respect to the cut $A|BC$ (and the other terms are defined similarly). If $\rho_{ABC}$ cannot be written in this form, we say that it is GTE.

We then show that the three qubit state $\rho_{ABC}^{(3)}$ is GTE by invoking the following sufficient criterion of GTE~\cite{GS10}:
\begin{equation}
|\rho_{1,8}|> \sqrt{\rho_{2,2}\rho_{7,7}}+\sqrt{\rho_{3,3}\rho_{6,6}}+\sqrt{\rho_{4,4}\rho_{5,5}},
\label{GScrit}
\end{equation}
where $\rho_{i,j}$ denotes the $(i,j)$ entry of a three-qubit density matrix $\rho$, where the standard product basis $\{\ket{000},\ket{001},\ldots,\ket{111}\}$ is assumed. Since for our particular state~$\rho_{ABC}^{(3)}$,  $|\rho_{1,8}|= 0.0621$ and the right-hand-side of (\ref{GScrit}) is $0.0588$, the criterion is readily satisfied. Therefore our state is GTE, as reported. 

Note also that we could not detect GTE for either $\rho_{ABC}^{(1)}$ or $\rho_{ABC}^{(2)}$ using the criteria of Ref.~\cite{GS10}. However, the numerical method of Ref.~\cite{JMG11} can detect GTE in these states as well. We then leave it as an open problem whether there exist three-party cyclically steerable states which are not GTE. In this respect, it is worth noting that for higher dimensional tripartite systems there exist biseparable states (i.~e. states that can be written in the form~(\ref{bisep})) in which all reduced bipartite systems are entangled. Such an example was presented in Ref.~\cite{persistency}. It remains to be proven, however, whether these states exhibit cyclic EPR steering.

\emph{Appendix D: Cyclic steering with POVMs and higher dimensional states.}---Here we show that if we do not restrict the local dimension to a qubit, we can strengthen our findings by generalizing cyclic EPR steering from projective measurements to general POVM's. To this end, we consider another construction, where the tripartite state is given by
\begin{equation}
\rho_{\tilde{A}\tilde{B}\tilde{C}}=\rho_{AB}\otimes\rho_{B'C}\otimes\rho_{C'A'},
\label{rho_abc}
\end{equation}
where the subsystems $\tilde{A}=AA'$, $\tilde{B}=BB'$, $\tilde{C}=CC'$ belong to Alice, Bob and Charlie, respectively. Let us further distribute the same states between the parties, that is, $\rho_{AB}=\rho_{B'C}=\rho_{C'A'}$. Furthermore, let $\rho_{AB}\in\mathbb{C}^{d_A}\otimes\mathbb{C}^{d_B}$ for some $d_A$ and $d_B$ be such that Alice can steer Bob's state using POVM measurements, but not the other way around (i.e. the state is one-way steerable for the most general POVMs). The existence of such a bipartite state with $d_A=2$ and $d_B=3$ was proven in Ref.~\cite{marco15}. More recently, the required dimension has been further reduced to $d_A=d_B=2$ in Ref.~\cite{bowles16}. In this case, the $(d\times d\times d)$-dimensional state $\rho_{\tilde{A}\tilde{B}\tilde{C}}$ with $d=d_A\times d_B=4$ exhibits the desired cyclic EPR steering property $A\to B$, $B\to C$, and $C\to A$. 

Indeed, in order for Alice to steer Bob (and also for Bob to steer Charlie and Charlie to steer Alice), the respective parties perform measurements on their entangled parts of subsystems.

On the other hand, the unsteerability property in the opposite direction follows from the fact that the composite state $\rho_{\tilde{A}\tilde{B}\tilde{C}}$ in Eq.~(\ref{rho_abc}) is product across the bipartitions $A|A'$, $B|B'$ and $C|C'$. Hence we have the reduced state $\rho_{\tilde{A}\tilde{B}}=\operatorname{tr}_{\tilde{C}}\rho_{\tilde{A}\tilde{B}\tilde{C}}=\rho_{AB}\otimes\rho_{A'}\otimes\rho_{B'}$, 
where $\rho_{A'}$ and $\rho_{B'}$ are the reduced states of $\rho_{CA'}$ and $\rho_{B'C}$, respectively. Due to convexity, if the state $\rho_{AB}$ admits a LHS model from Bob to Alice for POVM measurements, then the state $\rho_{\tilde{A}\tilde{B}}$ still admits a LHS model for POVM measurements. 

As mentioned above, using the one-way steerable state $\rho_{AB}\in\mathbb{C}^2\otimes\mathbb{C}^2$ from Ref.~\cite{bowles16}, we have a three-party cyclic EPR steerable state~(\ref{rho_abc}) for POVM measurements with dimension $(4\times 4\times 4)$. We state it as an open problem to find a cyclic EPR steerable state for POVM's in lower dimensions. 

