## Appendix A Complementary Quantum Measures and Relations  
Quantum particles can explore simultaneously multiple paths in a quantum coherent superposition. In general, the \(n\) alternative paths could be explicitly described with the use of quantum histories \(\hat{Q}_{1}\), \(\hat{Q}_{2}\), \(\ldots\), \(\hat{Q}_{n}\), which are defined in history Hilbert space as tensor products of projection operators at multiple times, and the corresponding quantum probability amplitudes \(\psi_{1}\), \(\psi_{2}\), \(\ldots\), \(\psi_{n}\) propagating along each path can be experimentally measured in the form of complex-valued sequential weak values [32]. Without loss of generality, after normalization it may be assumed that all given paths form a complete set of quantum histories \(\sum_{i}|\psi_{i}|^{2}=1\). In the special case of \(n\)-slit interference setups, the sequential weak values reduce to ordinary weak values [24, 25] with a single intermediate time at the slits, between the initial emission from the particle source and the final detection at a sensitive screen. The quantum _coherence_\(\mathcal{C}\) exhibited by each quantum particle in multi-slit setups could be manifested in _visibility_\(\mathcal{V}\) of interference fringes and is constrained by complementary relations to path _predictability_\(\mathcal{P}\) or path _distinguishability_\(\mathcal{D}\), the latter demanding quantum entanglement with external path measuring devises. **Definition 1**. : _Path distinguishability \(\mathcal{D}\) is defined with the use of external entanglement with measuring devices inserted on the paths. For the case of \(n\) paths, the distinguishability is [33, 34, 35, 36]_  
\[\mathcal{D}=\sqrt{1-\left(\frac{1}{n-1}\sum_{i\
eq j}|\psi_{i}||\psi_{j}|| \langle d_{i}|d_{j}\rangle|\right)^{2}} \tag{1}\]  
_where different device states \(|d_{i}\rangle\), \(|d_{j}\rangle\) are normalized but not necessarily orthogonal [37]. For orthogonal states \(\langle d_{i}|d_{j}\rangle=0\), the path distinguishability is maximal, \(\mathcal{D}=1\)._  
_For two equally likely paths, the path distinguishability reduces to_  
\[\mathcal{D}=\sqrt{1-|\langle d_{1}|d_{2}\rangle|^{2}} \tag{2}\]  
**Definition 2**. : _Path predictability \(\mathcal{P}\) is defined without the need of external entanglement. The idea is that if two paths have different probabilities, one could predict which of the two paths has been taken with a success which exceeds a 50\(\%\) random guess. For \(n\) paths, the predictability is [38]_  
\[\mathcal{P}=\sqrt{1-\left(\frac{1}{n-1}\sum_{i\
eq j}|\psi_{i}||\psi_{j}| \right)^{2}} \tag{3}\]  
_For two paths, the path predictability becomes [2]_  
\[\mathcal{P}=\left|\left|\psi_{1}\right|^{2}-|\psi_{2}|^{2}\right| \tag{4}\]  
Proof. : Indeed, squaring both sides of (3) and using \(\sum_{i}|\psi_{i}|^{2}=1\), we obtain  
\[\mathcal{P}^{2} = 1^{2}-(2|\psi_{1}||\psi_{2}|)^{2}=\left(|\psi_{1}|^{2}+|\psi_{2} |^{2}\right)^{2}-4|\psi_{1}|^{2}|\psi_{2}|^{2}\] \[= |\psi_{1}|^{4}+2|\psi_{1}|^{2}|\psi_{2}|^{2}+|\psi_{2}|^{4}-4|\psi _{1}|^{2}|\psi_{2}|^{2}\] \[= \left(|\psi_{1}|^{2}-|\psi_{2}|^{2}\right)^{2}\]  
Taking the square root on both sides gives (4). **Definition 3**. : _Path coherence \(\mathcal{C}\) quantifies the quantum interference between different paths [35]_  
\[\mathcal{C}=\frac{1}{n-1}\sum_{i\
eq j}|\psi_{i}||\psi_{j}||\langle d_{i}|d_{j}\rangle| \tag{5}\]_In the absence of any path detectors, \(\langle d_{i}|d_{j}\rangle=1\), the coherence reduces to_  
\[\mathcal{C}=\frac{1}{n-1}\sum_{i\
eq j}|\psi_{i}||\psi_{j}| \tag{10}\]  
**Definition 4**. : _Fringe visibility \(\mathcal{V}\) quantifies the contrast in observed interferometric distribution patterns and is given by_  
\[\mathcal{V}=\frac{I_{\max}-I_{\min}}{I_{\max}+I_{\min}} \tag{11}\]  
_For two paths, the fringe visibility reduces to coherence_  
\[\mathcal{D}=\sqrt{1-|\langle d_{1}|d_{2}\rangle|^{2}} \tag{2}\]0  
Proof.